Currently, Battlefield 3 does not support cross-platform multiplayer.
According to this article, which quotes the studio head, Robert Bach, there are numerous differences between the PC and Console versions of Battlefield 3:
Limitations caused by hardware requirements causes the maximum number of players in a game to vary between versions:
The biggest difference between the PC and console version of Battlefield 3 is that we have 64 players on the PC and 24 players maximum on console...We tried to get more players in [to the console version] but then you need to scale down all the graphics, scale down all the destructibility, and sometimes you need to scale down all the map sizes
Alterations were also done on the maps themselves to make them work better for smaller numbers of players:
When we say they are smaller, it's not that we have cut them in half. But we tried to compact them slightly to keep the action up.
Judging by these differences, and the licensing/requirements mentioned by Stof, it is extremely unlikely that there will be support for such a feature.
On a side note, having such a feature would be a huge win for this game franchise, which is in constant competition with the folks over at Call of Duty, and I'd hope that any game promoter/marketer would have immediately jumped on it as a major selling point if it was a plausible future enhancement.
This is kind of hard to answer because it's not exactly clear what the question is.
There are two primary game modes in Battlefield 3.
Conquest
Capture and hold points A-E. To capture stand near a point. The more people standing near the point, the faster it captures or uncaptures. The more points you hold, the faster the opposing team's lives go down over time. When one of the team's lives gets to zero, they lose.
Rush
One team plants a bomb on objectives A and B, the other team tries to prevent/defuse. Game is over when the attacking team runs out of lives (displayed in the lower left) or all the objectives are destroyed.
There's also deathmatch and team deathmatch, everyone hopefully knows what those are by now...
In general, this is my advice to new Battlefield 3 players:
Spot everything, all the time. The spot key (or button; it's q on PC, back on Xbox, and select on PS3) lets everyone else on your team know where the bad guys and vehicles are. Spot spot spot spot! If you do nothing else for your team, please spot baddies! Always! This should be instinctive.
Don't ever go anywhere alone. Buddy up. Find your squadmates (in green) and follow them around. If you are not in a squad, join one using the SQUAD tab on your loadout screen. If your squad is filled with idiots, leave the squad and find another. Try to pick squads with players of higher rank so they aren't beginners and at least in theory know what they are doing.
Always spawn on your squadmates when possible. Choose your spawn point via the DEPLOY tab on your loadout screen, which will include each of your squadmates, provided they are still alive. If you cannot spawn on your squadmates, spawn at a nearby point -- perhaps not too close, if you are getting killed a lot, to give you time to approach the action without appearing in the middle of it.
Move from cover to cover. Don't just run down the middle of a field or street. Pick one piece of cover, then pick a path to the next piece of cover toward your objective. Repeat until you arrive.
Play the objective. Those big visible A-E letters are what everyone is fighting over. Just shooting bad guys is not enough. If you want to win, help your team achieve the objectives of the map.
Half of the battle is knowing the map. It helps a lot to know the common entry and exit points on the map, because otherwise it feels random -- like anyone can be coming from anywhere! Try to stick to one map for a bit and learn it reasonably well, before moving on to the next "never seen this before, no idea where anyone is going" map.
Use a microphone. Talking to your teammates is very helpful. Try and get in a squad where others are talking, and talk to them. This is a great way to improve your game - you'll become more aware, work better together, and maybe even pick up tips from more experienced players.
Of course there is tons more, but that's the essentials.
Best Answer
This has changed a bit for final.
This is a global change for all multiplayer modes. Basically they are moving a bit more towards the Call of Duty: Modern Warfare behavior.