turn order i think you already understand. whoever is fastest, goes first. can't break it down any other way.
what happens next depends on what the first champ to go does. is it a turn meter manipulator like arbiter or high katune or apothecary or seeker or one of the many others? then they boost the rest of their teams turn meters effectively making them "Faster" as well. If it's a deacon or someone, they might not only boost their teams meter, but also reduce the other teams meter, making it so the rest of their team has an even better chance at going without getting "cut" in.
some champs have passives that boost their tm every turn, masteries come into play, etc... all kinds of stuff effect that pesky little TM.
watch for passives. they could determine who goes next or doesn't.
valkrie is a good example, she has a passive that boosts her TM everytime the enemy team gets a buff, so she often gets to cut in.
tormin has a passive that has a chance to freeze the enemy when they take a buff, making it so random people on the enemy team might get frozen, adding a bit of crowd control to teams. tormin also has
watch for provoke champs, they're also great crowd control and can screw the enemy teams strat up.
tormin also has a provoke, if he goes before your team, you might end up provoked. what that means is your champ can't do anything but attack that champ with your a1 until that provoke goes away or the enemy champ dies. Umbral Enchantress (void epic) is a particularly good trap, she places an unkillable on herself for 2 turns and places a provoke on the enemy team, completely locking them up cause all they can do is attack an unkillable champ while the rest of her team does work on the enemy team.
if you're looking for tips on how to help your own team without the need for speed, a shield set on a high HP champ can protect you against an initial nuke from a weak team.
retaliation or avenging sets can be used on champs with an AOE first ability. when they get hit, they have a chance at counter attacking so they'll hit the entire enemy team. avenging is probably best for this in higher levels of arena since most hits will crit anyways.
if you want a tanky champ that will survive the initial barrage, a stalwart set along with masteries will mitigate a lot of the damage.
frost sets can randomly freeze attackers as well, adding some annoying crowd control.
so knowing all of that, and knowing that building a speed team has been drilled into your head by this point probably (high katune, buffer, debuffer, nuker), what do you want to do?
provoke teams are very effective, but the provoker needs to be fast and (if it's someone besides umbral), tanky.
there are go second teams where you can have a really tanky cardinal built, she lives, the rest of your team dies, then she revives them all with full tm's.
there's crowd control teams (much like provoke) that just depend on stun, frost, etc. proccing. really they're trying to be annoying as possible.
there's also counter attack teams, make use of the avenging gear paired with someone in a shield set.
and as you've probably guessed, looking through all of those, there's a lot of rng involved if going against an equal opponent. if its a speed race, they might be faster. if you plan on going second, your gear might not proc, or they might nuke you into oblivion or cc your team themselves before you get a chance to go.
there's also a ton of other little random tricks in the arena. siphi will cleanse a rotos for example, so if he's stunned or frozen he'll still go. nekhret the great will throw an ally protection on the weakest champ, so a glass cannon might survive and get their nuke off. serris will make your sheild set useless cause she'll strip it right off. vogoth will absorb a ton of damage for the rest of the team, giving them a better chance to survive.
despite what a lot of people think, this game isn't about pulling the meta champs. yea maybe in platinum arena you need a stacked roster and stacked gear, but for the rest of us, it's about figuring out what roles need to be filled and what champs you have that can fill those roles. i already gave the example of turn meter manipulators, that's a roll that needs to be filled for a speed team, and there's a dozen or more champs that can do it. if you build a provoke team, there's almost 2 dozen champs that can do an aoe provoke, and some are rares or even uncommon. you gotta look at your roster, what you have available, and fit it all together, and this goes for every single part of the game, not just arena.
occasionally, you WON'T have the champs needed, so you gotta find another way or hope to get some lucky pulls someday, or maybe a fusion will come a long with the perfect epics or lego for you. i still don't have a counter attack champ so i could never do a CA clanboss team, but i eventually built an unkillable team with other champs. work with what you got.
also the difference between a level 50 and 60 char is night and day for most arena champs, so once you get your arena team picked out, rank them up, ascend them, and get full masteries on them. it's a grind, but it's also a must. like on a speed team you can get away with a level 50 turn meter manipulator, maybe a level 50 buffer/debuffer, but your nuker and any tank needs to be 60.