I go through the game avoiding as many random battles as possible (with the exception of Magus' castle -- I make sure to get Glenn Heal before fighting him), but fight any random battles I run into. As far as sidequests, I attempted a run through with Lucca and Ayla while doing all the sidequests, but it was way too easy. So now I've resigned to doing just Crono's revival, the Rainbow Shell (Prism Helms pls), and one run through the Black Omen. I usually am level 9 by the first trip to the End of Time, level 29 before challenging Giga Gaia, and level 43 for the fight with Lavos in my new, shorter playthroughs.

Since I planned to do 3 different playthroughs, I figured I'd save myself some time by using the same base file up to the point where I have to decide between the 3 character sets in question. I used Robo and Lucca instead of Marle and Lucca (as I have done for all of my playthroughs ever) for Heckran Cave, Zombor, and Masamune's Mountain. That turned out much better than I thought. Robo's Rocket Roll/Supersonic Spin clears Heckran Cave and most other random battles with much greater ease than I'm used to. The fight with Zombor was about the same as usual. Masamune was also about the same. The lack of Aura Whirl didn't matter nearly as much as I thought it would. I made my base save file before talking with Ayla to start the sake drinking.

By the way, Crono is assumed as a party member, even for late game, because 1)He's a good character and 2)You invest so much time and TP into him, it'd be a waste not to use him. Besides, being forced to use him for so long doesnt' really let you raise a third member.

So far, this is the only playthrough I've completed. I decided to give my Power Tabs/Capsules to Crono, and split my Magic and Speed Tabs between Lucca and Robo. I also pretty much had to wear Speed Belts on them for most of the game, those slow whores.
Nizbel: I killed a few frogs (two runs through the forest) for trading materials (Sword, Gun, Arm, and Ruby Vests) before talking to Ayla for the party. After the party I decided to take Robo to Nizbel because he's my only healer and I've always taken Lucca in the past. Nizbel was pretty easy, but then he always is. Stone -> Sword -> Flashbacks -> Water for Glenn -> Mountain Cutting -> Magus' castle.
Magus: Once again, I decided to take Robo for this one. Turns out Rocket Roll/Supersonic Spin is amazing for clearing out most of the enemies in the castle. I'd say it's even better than Ice Water. This combination also allows for Triple Attack, which made quick work of Slash, but Flea was still annoying as usual. Yadda Yadda got Heal, fought Magus with Lucca instead of Robo. With Glenn spamming Heal, this fight is cake. Lucca was very helpful. Finally able to use my true team (Lucca and Robo) for the climb up the mountain. Then more forced character usage >.>
Black Tyrano: Lucca the whole way. No problems at all. This is a pretty good team for climbing the tower, and a very good team for killing the Tyrano. Nizbel II always has the same tactics: spam Volt Bite and heal with the third character. Tyrano is made much easier by Ruby Vests. From this point, I can use my intended team until Crono decides to liquefy himself.
Giga Gaia: I got somewhat lucky against the 6 Nus, but if I wanted to play it safe I would have spammed Heal Beam. Ruby Vests and Robo Tackle + Fire Sword (in that order) made Golem a joke. I decided to get nearly everything I could get with my shiny new pendant. The important things are the Gold Stud, Magic Ring, Tough Ring, Speed Ring, colored Vests, and some various tabs. I chose the Swallow over the Safe Helm because it would end up being the sword I use to and through the end of the game, for lack of a Rainbow. The Beasts and Mud Imp weren't much of a problem. I mostly used Fire Tackle and Lightning 2. I love Gold Rocks! They die in one Swallow crit (which it seems to do often), and they get me all the abilities I could ever want. I ended up at Giga Gaia with Confuse/Frenzy, Uzi Punch, and Megaton Bomb. I didn't have any particularly great techs for dealing with Giga, but he still went down pretty quickly with the previously mentioned abilities. Colored vests are helpful.
Golem Twins: Dalton isn't worth mentioning. Sprinted through most of the underwater Omen. The colored eyes were somewhat annoying, but I only ended up having to kill the blue ones with physical attacks. Fire 2 and Lit 2 were used to clear out most of the mobs. Fire Zone is surprisingly weak. The Twins themselves were incredibly easy with the same strategy as before. Fire Vests -> Fire 2 -> Frenzy -> Double Bomb -> Win. I think Fire is their worst element. Crono gets vaporized, lots of story, etc.
Dalton: I took Ayla as my third companion on the Blackbird because she rocks. I was (luckily) able to kill 3 turrets before they disappeared. Their avoid is so much higher than the Gold Rocks. Golem Boss died before he even got to 1, thanks to Double Bomb. Dalton was lol.
Crono's Revival: After sucking at the Mime game (I blazed through the instructions and didn't see which button was Laugh), I headed up to fight the three Lavos' Spawns. Fire Tackle and Cat Attack made quick work of the three of them. In hindsight, this didn't deserve it's own bolded section. They're crazy easy.
Zeal: You may be wondering why I did this before the Rainbow Shell. Well, I thought I could beat the game without the Shell, but I came out of the Omen at level 40 with nowhere near enough magic defense to make Lavos any kind of beatable, so I went back to get the shell (because of the helms, 9 mdef each). The Omen was pretty hard because I didn't have my usual status immunity and I came in at a pretty low level. This team doesn't have any particularly good screen clearing Double Techs, so I resigned myself to putting a Gold Stud on Lucca and spamming Flare (I didn't have Luminaire until halfway through the Omen). Also, yay for stealable Speed Tabs! Mutants were easy, Greater Lavos spawn was easy, Zeal I was easy, Mammon Machine was pretty easy. Actually, I think I did worse against the Tubsters than any boss in that area. Zeal's Face was incredibly easy thanks to having Robo lead the team. I don't think any team could kill her faster than Double Bomb + Frenzy.
Yakra XVIII: At this point I was a bit too strong for this area, so the whole thing was a cakewalk. The Rust Tyrano took a little time to kill, but no problems otherwise. Trials are slow and annoying. Chancellor XVIII can rot in his box for all I care.
Lavos: Robo was still OHKO'd by his starting attack. After Life+Megalixer, I decided my plan for the arms was Flare, Luminaire, and Shock/Heal Beam. The head died to Frenzy, Uzi, and Flare. By the way, he has some sort of eye-beam that's a bit stronger than his initial armageddon, so Life was used a few more times. Poor Robo. His last form was actually pretty hard for this team. The left bit died immediately to Frenzy+Uzi Punch, but then Lucca died shortly after to a giant f'ucking rock. The center bit was annoyingly hard to kill. Once the core put down its defenses, I spammed Luminaire and Flare while I could, but I had to put Robo on healing duty until the center bit died. Once it did, I was finally able to barrage the core with a whole lot of Double Bombs. Even though he revived the bits, I was able to take him down before they were able to do any real damage to me. Life was used more often than I made it sound, by the way.

Overall Rating: 8/10
Unfortunately, a lot of this team's best power is wholly situational. More specifically, only if the enemy is really close to Robo. In this case, you can use Max Cyclone early game and Double Bomb late game. If Double Bomb is usable, their power is pretty much unrivaled. However, they don't really have much going for them if the enemy is far away. Double Volt is only barely stronger than Luminaire due to the horrible magic stats of the two characters involved, so Flare is your main skill for taking out the screen. Even worse, Flare isn't gotten until fairly late-game, and it's not nearly as strong as some other screen-wide attacks. Most notibly Twister, Inferno, and Antipode 3. The difference in power could easily mean the difference between a screen of dead enemies, and a screen of mostly-dead enemies. Still, this team is better than almost any team involving Glenn or Marle, so it still gets a generally good score. I just think the other two teams will end up better.

I'm mostly through this playthrough. I'm just about to revive Crono. I decided to give the Power Tabs to Ayla. At first I decided to give Ayla the Magic tabs too, but after giving her 5 of them I thought I'd give them to Crono instead. Luminaire has a crazy multiplier, so I figure they'd benefit him more than the others the way my team is set up.
Nizbel: I did my trading material farming after the party, this time. I used Robo as my tertiary character, obviously. It went just as well as the above playthrough. Sidenote: Rocket Roll is pretty amazing for dealing with those little peon Reptites.
Magus: I used Robo the whole time. It went really well (as noted before) for the Slash fight, but Magus was a totally different story this time. Magus has incredibly high defense for the first 3333 HP, and if you look at my team, you'll notice none of them have very good magic stats or attacks. As such, the first half of the fight went very slowly. Cure Beam was pretty much spammed, and I used Sword Stream or Lightning when I could. Triple Attack made the second half of the fight very quick and easy, though.
Black Tyrano: At this point, my team is more or less set. I put Berserk Rings on Ayla and Robo for most of the climb to the Tyrano and everything died very quickly. Nizbel II died to Volt Bite, again. The Tyrano himself was actually a huge problem, though. Azala, once again, is a character with low mdef and high phys def, so there was little more I could do than use Lightning and physical attacks. He even almost finished a countdown before I was able to kill Azala. Tyrano was killed with Volt Bite and Robo Tackle. Meh.
Giga Gaia: I knew the 6 Nus would be hard unless I played it safe, so I spammed my newly-acquired Lightning 2, put a Berserker Ring on Ayla, and spammed Heal Beam. No one died. Golem was a huge problem though. Since my team lacked fire, I decided to attack him with lightning, which it turns out is much stronger than Fire. Also, he sometimes does a weak, single-target "Flame" with Fire that's just great. With lightning he just kept using a full-screen strong attack that I had no decent armor to block. It went very slowly (with lots of Revive's and Heal Beams), but I was able to down him. I got all the same treasures (Swallow too) as the last playthrough and headed to the Imp. Even though I took quite a lot of damage from counter attacks, spamming Falcon Strike was a pretty quick victory. Gold Rocks died and Giga Gaia was reached. Both arms died in a single newly-acquired Twister (lol), so I didn't take any damage aside from their pre-emptive attacks. I think that's my best Giga fight ever on a first playthrough.
Golem Twins: Dalton died in a single Beast Toss. I put a Gold Stud on Ayla and wiped out most of the enemies with a single Twister (except the Scouters -- they died in a single Falcon Strike). The Twins were horrible though. I decided to use Shadow this time (which, it turns out, isn't really any different from lightning), and spent a very long time killing them. Beast Toss wasn't an option because of the horrid physical attacks the Golems have, so I ended up killing them with several Twisters.
Dalton: I took Glenn as my third character for some reason. 3 cannons died again, and the Golem Boss died incredibly quickly to Beast Toss. Dalton died in one Beast Toss again. He's funny.

This is as far as I am now. I'm about to go revive Crono.

(Tenative) Overall Rating: 9/10
This team is great for clearing out the little guys. You can spam physical attacks when you feel like conserving MP since their physical is high, and you can clear the screen pretty damn well mid and late game with Twister. This team's main weakness comes with a few specific bosses. Mainly, those with high physical defense early and mid-game and the Golem Twins. If you weren't rushing through the game like I was and you knew enough, you could learn Charm before heading to Mount Woe and get some Rainbow Helms from the local Mud Beasts (which resist Shadow) to plow through the Twins. Still, in situations like the Greater Lavos Spawn, Zeal's Face, Lavos and... most bosses that can't be Beast Tossed, Robo is a 5th wheel. All he has to offer in those situations is mediocre healing, mediocre single-target physical attacks, or mediocre full-screen magic attacks. No doubt this team is very powerful in many (especially non-boss) situations, but I think we can do better.

As I said before, I ran through with this team first on a ROM, albeit with a slightly more item-heavy route and all the side-quests done. However if you pump Ayla with all your Magic Tabs, Inferno becomes a force to be reckoned with. Luminaire and Inferno make quick work of any boss with multiple targets or random enemies, or you can switch to Blaze Kick and Frenzy for situations like Zeal's Face. As a sidenote, Kiss also heals for a pretty absurd amount when your Ayla has more than double of the magic she was intended to have. I'm pretty confident this will be my choice for the best team because they can adapt to the situation much better than the above teams while still being pretty damn powerful.