I've got three games down now, each as a different Civ. Even before going into it, I knew my first game would HAVE to be as Caesar, so I'll start with him...
ROME - Rome is a Civ that, I found, encourages large, developed Empires. It's a shame you can't build a Courthouse in Rome, but GoR is still fantastic without it. The UA definitely requires you to put a little more planning into your build order, as well as planning for other cities to take over Wonder-building duties, but as long as you make sure to build needed buildings in Rome first, you will soon find yourself with an excellent core of cities that you can either have specialized to certain tasks, or all working to be 'jack-of-all-trade' cities, as the terrain permits. I would highly recommend going for Liberty as your first Policy when playing as Rome, and push out towards multiple Luxury sources as quickly as possible.
Rome's UU's further reinforce a rush-then-turtle approach. I push for Ironworking as quickly as I can afford, after grabbing Calendar (if the associated Luxuries are near by). Playing Rome, you want to find your Iron sources as quickly as possible, and aggressively settle or, if needed, conquer for them. If you can, provoke a war by settling close, then push the advantage of your Legions and Ballista for as long as you can. Because of the Ballista's power and bonuses v. cities, I found I only ever needed one with my Legions, and never bothered to upgrade them until Cannons. Once you've secured a large empire, turtle up and develop your holding with Glory of Rome. From there, you can push all you want for whatever Victory you desire (though a large empire might make Cultural a little more difficult). My personal favourite SP's when playing Rome is Liberty, Piety and then Order.
FRANCE - Not quite finished this game yet, but the UA definitely allows for aggressive early land grabs and SP-related development. The timing of their UU's can make defending such expansion a little more difficult though I found, especially since I started next to Babylon on one side, and Greece on the other. Of course, your mileage may vary. For SP's, I went with some of Tradition to take advantage of Paris's quickly-booming borders, and Unlocking Liberty only for the reduced cost of Settlers. I found myself spreading my cities out a bit more with France, since I knew their borders would expand towards tiles a little more quickly, so Oligarchy for defence, and starting Honour for the Barbarian warning is good.
Their UU's I'm honestly not a huge fan of though. Musketeers are good for what they replace, but overall, they require a little more forethought. You can't upgrade your Pikes or your swordsmen to them, and they have a very short lifespan before Rifling comes in (and lets you upgrade your other Ancient+ units). I only really built and used them to get the Steam achievement. Foreign Legion are more impressive for sure, but force a very aggressive campaign. On the defence, they're no better than regular infantry, which can really hurt if you're caught by a counter attack right after taking an enemy city.
PERSIA - Originally I started a game with them because my buddy wanted to try Greece, and I was trying to be funny. The result was actually quite enjoyable! Darius is a leader that I think a lot of people have written off because of the general 'meh!' attitude towards the current Happiness mechanics, but I think the power of Golden Ages is really underestimated. On top of the usual powerful bonuses to production, your military suddenly becomes a lot more mobile. If you plan your campaign timing to start with a Golden Age, this can allow you a brutal first strike against an enemy that never knew you were massing forces in the first place, and the combat buff will help make sure you can exploit the gaps you open in his lines. Darius also has some of the best synergy of any Leader between his UA and his UB. A Bank that puts more Happiness towards my awesome Golden Ages? Yes please! Also, I would call Piety a must-have Social Policy when playing Persia.
Their UU isn't half-bad either. Especially on larger maps, they'll allow for your exploring Spearmen to go after Barbarian camps a little more recklessly than normal, as the healing time is slashed in half out in no-mans land. It can also allow for more aggressive Ancient/early Classical-era campaigns, as you won't have to stop as long before moving on. This can also result in a more experienced army, as you won't be feeling quite as pressured into using promotions for insta-heals.
Anyways, that's my little bit of first-impressions, after a game as each of the above. I think my next game, I'm going to try Songhai for their Cultural boon of a UB. And please, feel free to critique/comment on my analysis!