Rogue/Fighter works at so many levels and combinations. It's really unique as a multiclass in that way
More important than the finished lvl 20 build is the journey along the way
Rush to 3d6 sneak and Uncanny dodge with 5 levels of rogue? Or rush to 2 attack rolls (2 chances to get a sneak attack off) and a 2nd ASI/feat with 6 levels of fighter? Or go slow and secure Action Surge and Cunning Action before your big lvl 5 class features. I lean towards getting 5 levels rogue or 6 levels fighter and then switching over
Your bonus action should go to Cunning Action usually. Anytime you get one incredibly powerful bonus action outlet, it's probably best to just roll with it. Getting crossbow expert or some other bonus action trick is probably not the best idea here. Having Tasha's Rogue feature, Steady Aim is also a game changer since now all of a sudden you don't have to play hide and seek or rely on flanking house rules to guarantee those sneak attacks
Fighting style? Bows and archery is great. My lvl 10 Arcane Trickster single-class rogue just picked up archery style for his level 10 feat (Feat from Tasha's) just to really guarantee those hits and sneak attacks
Starting fighter is *probably* better just for the option of using heavy armor (can maybe find mithril versions for being sneaky...) and getting CON save proficiency, which is probably better than DEX save proficiency
Also I can't state strongly enough how good a Battle Master is. The versatility with combat manuevers is just amazing. Turn misses into hits, turn hits into big hits. Grant disadvantage, trip enemies, or even take free movement. Battle Master is a beastly subclass and flavorful as well. Champion is unfortunately quite a weak subclass except with certain critical-hit centric builds. So stacking critical hit riders, being a half-orc, etc. will open good synergy. But most builds just don't get much out of the Champion. Psi Warrior is another nice subclass since it is also INT based like Arcane Trickster. It will give you competition for your Uncanny Dodge reaction, but you know what, sometimes you aren't getting hit and your allies are and you can block that damage also, which is fine
Two final details about Arcane Trickster.
1) Always get Find Familiar. It lets you use the help action *every round* as long as it lives.
2) Always get at least one cantrip melee spell like Booming Blade. So lets say you only have one attack roll. Why not make it a Booming Blade? It gives you at least 1d8 free damage on a hit (+2d8 if they move) starting at level 5. It scales up again to +2d8 (+3d8 more if they move) at level 11. Just a lot of extra damage for zero downside
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