No-one can tell you what to buy, but we can tell you about the games.
Bioshock 1, and to a lesser extent 2, have rather... broken gameplay. In one, a lot of it revolves around having to right click to swap between weapons and plasmids, as opposed to just being able to use one from left click, and one from right click. Bioshock two fixes this, but both have the problem of the insta-win package: Lightning and Shotgun. Lighting is the first plasmid you get in both, and the shotgun you get in the second area of Bioshock, and I believe the third area of Bioshock 2, so you get it early on and can just keep making it more powerful.
Many of the games aspects can be a grind, and some are semi-required to get through it without dying a lot. Things like searching for healthkits and eve hypos in both can be interesting, or can be a grind, and searching for new plasmids and taking pictures of splicers for bonuses (MASSIVELY improved in 2) is a massive grind. Not to mention killing Big Daddies in both can get more annoying than fun after you figure out a formula, and just have to wait for them to take enough bullets to die.
Most of the campaign in Bioshock 1 & 2 is strung together fetch quests. Hell, I think that's all of the campaign in one, and far less prominent in 2. As far as the stories go, both are great. One is far less personal, but has what I believe to be the greatest twist of all time. If you don't know it DO NOT LOOK IT UP. And nobody tell him. It is far more powerful when you don't expect it. 2 has no real twist per se, but is a far more personal story, and has an intro that I just love.
Bioshock 1 has no multiplayer, but Bioshock 2 does, even though it is largely dead and horribly laggy (Not even in a game and I was getting 150+ ping... in the menus. I am used to 20 or so ping to servers in game), but it can be fun and kinda ties in with the story too.
Battlefield 3 is much the opposite. Tight gameplay, and very fun. It is not a grind after the first bit (Where you're fighting to get enough points for either a new gun or some form of attachment. The worst part of the game IMO, but it gets far better once you pass that), and matches can get crazy at times (Go on a 64 player Metro Conquest. You will not leave without at least 5 deaths [Even if that does include teammates reviving you. The exception is if you join the game late]). The 4 classes provide different roles on the battlefield, and each battlefield has opportunities for those roles to be used to great effect. Get the Back to Karkand DLC for it and you'll get IMO the best maps the game has to offer, as well as even better destruction than the default game whilst on those maps.
The campaign in Battlefield isn't as bad as many say, but its nothing great. Its a large resounding meh. It is mostly just a 'Run forward and shoot things' type of campaign, with quicktime events reasonably often, and a railshooter section for the Jet level (You get no control over the jet, you just lock on then click to fire and deploy 'invulnerability' flares. It also has no storyline significance at all. Its whole purpose is to look pretty - which it does - but its a crap level). Don't expect any great twist you didn't see coming from the start, though you could do worse in a campaign. More meh than anything else.
Multiplayer is the focus of BF3, mostly competitive. The Co-op is alright, and really, really forgiving, but it goes through the paces and will get you to experience different roles that you may undertake on the battlefield that you didn't do in the campaign (AKA: Flying an attack helicopter) and get you some experience in those areas without the high risk of competitive multiplayer. Also, there is one level where you literally play gangsta at the end, and one player drives a car out of a carpark whilst the other shoots enemies out the window with an SMG and Shotgun with pimp music playing.
The competitive multiplayer's first five matches or so are some of the worst in the game. I'd personally recommend going on a 64 player Metro Conquest server for them as, though you'll die a lot, you would have anyway and you are at least pretty much guaranteed some kills. Once that's done though, and you get your first couple of unlocks, the game becomes much better. People with better weapons will keep killing you, but if you learn the maps you can outflank them and kill them easy enough. Vehicles provide a means to semi-safely battle the enemy, but definitely get the smoke upgrade and keep it for all vehicles. Them Javelins be evil. Far more of a reflex based game than Bioshock, but still very entertaining.
So, if you don't mind clunky gameplay, and want a great story - get the Bioshock series.
If you don't mind a meh story, but want tight online gameplay that is fast paced - get BF3
