I'm extremely impressed by the first battle scene in the show, where instead of a bunch of soldiers fighting you can actually see the extreme discipline of the Roman legions. Looks like the producers did their research in regards to the army.
My question is how accurate was their depiction of city life in Rome itself?
Alrighty, first off - I'm going to preface this with a couple of disclaimers. Or a few. Whatever :P
I LIKED Rome. What I say below doesn't change that.
The show is more accurate than most movies that are "based on history.
Obviously, there are spoilers. I've only seen most of the first season, and I'm basing this off of what I've seen. If you want to continue watching the show in blissful ignorance of the stuff that makes me go "Well that's not quite right...", then please, don't read any more!
Okay, now that the prelims are done with, I'm going to address the idea that the producers did their homework regarding the Roman army. Honestly, it's better than most (They're not wearing lorica segmentata, for one thing), but again, it's not quite right. I love how they showed the discipline of the Roman army, however, again, it was not quite right. For example, Pullo, when he breaks formation and strikes a superior officer, would have been executed on the spot. No questions. The Roman army had extraordinarily strict discipline and specific, harsh laws regarding their behaviour. Pullo should not have survived the first episode. Speaking of the first episode, the show depicted a battle - however, I'm not sure WHAT battle it was, even though they claim it was Alesia (By Vercingetorix being captured afterwards). The Battle of Alesia was a totally different beast - it was a desperate siege, with the besiegers themselves being besieged, and an incredible display of Julius Caesar's brilliance as a commander, as well as an indisputable example of Roman engineering at its finest. I didn't get that from the show :/
Then, one little nitpick that would have been a HUGE deal. Caesar (and others) rode through the Roman camp. Again - seems like a funny thing to nitpick, no? The thing is, no one, not even kings and commanders, were allowed to ride through the camp - it was considered extreme bad luck, and at least one Roman loss that I've read about was blamed partly on a couple of individuals riding through the Roman camp. Caesar, a man who publicly based his reputation on his fantastically good luck, would never ever have committed such a faux pas.
Oh right. Speaking of Caesar and faux pas-es. Another thing he would never have done is shown that much favour to the Thirteenth Legion over all of his others - his favoured legion was the Tenth, and he spent the vast majority of his time with that legion, as opposed to the others. The Tenth went with Caesar almost everywhere he went (The only exception being the one legion he took with him when he went to Rome, crossing the Rubicon - which WAS the Thirteenth). When Caesar fought the Alexandrian Civil War, the Thirteenth wasn't originally with him, as shown (poorly) in the show - that was the Sixth and the Twenty-Eighth. But they did get the armour right, and it's absolutely possible that they rotated their ranks like they showed in the show, even if we have no proof of it! :)
Now, on to non-army things. As /u/Tiako noted so well here, with broad strokes, the show is pretty solid. Showing street life is fantastic, showing the colours of the city, rather than the blase, stolid marble that we're used to is utterly gorgeous. But then there are the little things - and I'll just give you a couple of examples of those.
First of all, Caesar's...uh....sleeping around was rather well known. He'd slept with half of the married women in the Senate, it seems, and Servilia was Cato's (Who was FAR too old in the show, and would NOT have worn a black toga in the Senate) half-sister. Also, she had a daughter - who Caesar also slept with. Cato's wife? Yeah, Caesar slept with her too, I believe (He divorced her for sleeping around). Caesar's legionaries sang songs about how much Roman tax money was spent on Gallic women. My point to all of this is that Caesar's wife would not have been surprised at his propensity to bang everything with boobs and two legs. She certainly wasn't about to divorce him over it.
The one big one that ground my gears a bit - they got housing completely wrong. The vast majority of Romans would have lived in houses called insulae, which were essentially big tenement complexes. Rooms were small, shoddily built, and they were generally five stories tall or so. Contrast that with Lucius' lodgings - the room is relatively comfy, larger than you would get from one of these insulae, and, most of all, it had something that those tenements would NOT have had - a kitchen. Funny how we just take that bit for granted, eh?
And then, of course, the excessive incest and random sex between Servilia and Octavia, Octavia and Octavian, Atia being batshit loony, etc etc are completely ridiculous. If you have specific questions about what you're looking for, please, feel free to ask them :) The issue with this question is that it's extremely broad - but in general, this all sums up to NOT ALL THAT ACCURATE. Compared to other Hollybood butcheries? It's amazing. Compared to reality? Not all that true.
If you'd like an amazing overview of life in the Late Republic, feel free to check out Adrian Goldsworthy's Caesar: Life of a Colossus. If you'd like some other book recommendations, just let me know and I'll provide :)