versus mode is exactly what it sounds like. Everything in here is all about direct competition with others. You can choose either Quick Match or Ranked match for all gametypes where Quick Match will put you into a game as quickly as possible, using bots in place of players when necessary. Ranked will wait until every spot has been filled before beginning and will be more competitive in its matchmaking. There is actual a final category called casual which is strictly for people who haven't played Gears of War before. You can only access it if you have no achievements from the first two Gears of War games and for your first handful of matches in Gears 3, after that, you are kicked out and can't get back in.
Team Deathmatch
Each team starts with 15 respawns and once those are gone, people can't respawn. Whichever team is left standing at the end, wins. This might sound simple but it turns it into a very tense game where you don't spend a lot of time sitting around. Standard rules are first to two victories wins the match, which is important because if the game becomes lopsided and there's only one player left, if he can get the other team down to one player, it will become a sudden death match. They have 60 seconds to find each other; otherwise it's a draw and goes to another round. In this mode, you don't have to execute people and tactical thinkers are rewarded so I highly recommend going with the hammerburst and Gnasher for most maps.
A last note and a very important one. In this gametype, whenever the Locust queen or Prescott announces "Enemy Forces!" or something similar, it means that the spawns have swapped sides so it would likely be prudent to turn around.
Warzone
warzone is essentially the tail end of Team Deathmatch on its own. The match is the first to three rounds and I suggest you stick with the Hammerburst here. If you die, you're done for the rest of the round and have to wait for the next so it pays to be prudent. In this mode, you really have to play smart if you want to win. Make sure you plant smoke grenades behind you so you know if you're being snuck up on and make sure you spot enemies early and often.
Execution
Execution is a lot like Warzone. If you get killed, you're out of the game and it's the first to three rounds. On the other hand, the game is a far more aggressive style. You can't kill enemies from far away unless it's with a one hit kill weapon so power weapons and close range weapons are a premium in this mode. It's also highly recommended that you use meatshields in this game mode. since these matches can turn into close quarters firefights and you have to be on top of your enemy to kill them (likely not in cover since you've already downed them), it would generally benefit you to take them as a meatshield and protect yourself rather than kill them and try and get out. In this mode, both the sawed off and Lancers are far more viable. I would leave the Hammerburst on most maps and go for the regular Lancer and your choice of shotgun depending on your preference.
Capture the Leader
capture the leader is very different than the other game modes. It's essentially Capture the Flag where the flag is a person who shoots back at you and to capture it, you actually have to hold on to it for 30 seconds. The match is played to two.
For most eight of the ten people (the non-leaders), you have to balance offense and defense while keeping track of where both leaders are. You can always see where both leaders are so use this to your advantage. If the Leader is moving in a particular direction, try and come around behind him because it's likely that his team is moving with him. Eventually, someone must down the leader and take them as a meatshield.
Now this meatshield is not dead like in most other gametypes and can struggle. If they hit the B button, it will cause you stop for a moment (which is dangerous, as you'll see shortly). If you get downed, even from far away, you are likely to be dead because once the Leader is released, they'll get back up and be able to kill you.
Once you have the Leader, you must hold on to him for 30 seconds. You take slightly less damage while holding the leader but the Leader doesn't actually prevent any damage. If someone shoots the leader at close range with a shotgun, all of that damage is actually transferred to you and the leader will be released. Also, if the other team manages to hit you with a smoke grenade, it will immediately cause you to release the enemy Leader. This is usually the most effective method, especially if the leader is smart. If they wait for just the right moment, they can hit the B button, causing you to be unable to move and allowing the smoke grenade to land right at your feet.
As the leader, you are an interesting unit. You can't die in this match for any reason which can make you very dangerous, but you're also the objective so you have to balance when you engage and when you don't. You can also see everyone on your Tac/com, including the entire enemy team. If the Leader is a good communicator then he can convey this to his entire team and dominate the match.
King of the Hill
This is the most typical of gametypes. There is a hill on the map that appears as a colored ring. White is unoccupied, Blue is COG, and Red is Locust. To gain points, you must capture the hill by occupying it for a set period of time. When a team starts to claim the hill, you can see the icon in the upper left start to fill. It will fill faster if they have more teammates in the hill. Once the icon fills, they will gain one point per second whether they have someone in the hill or not until someone from the other team enters the hill. This is called a break and it immediately clears the hill to neutral. It also has the same effect as the hill is filling. Even the icon is nearly filled, it will empty to 0.
What all these means is that in KotH it pays to be aggressive. You want to get in that hill as quickly as possible and clear any enemies. The only time you don't want to be in the hill is when you've already claimed it and you're gaining points. In fact, at this point, it is somewhat beneficial to get out of the hill and set up a sort of perimeter around it to stop the enemy from getting in.
Because of the fast paced nature of KotH, I highly recommend close range weapons. It would not be over kill to go with the Retro and Sawed-off on most maps.
Wingman
Wingman is a first 15 points gametype and goes as many rounds as needed to get there. You gain these points by killing enemies. You get one point for every enemy you kill and one point for being the last team standing in a round. As for teams, there are four teams of two players each, and both players on a team are the same character. So Team Kantus will both be Kantuses. This is one of the most difficult gametypes to give advice on because what weapons you use and your tactics all really depend on your teammate.
The only solid advice I can give is to bring a friend. If you know the person you are playing with and communicate well then you stand a much better chance of winning. I would recommend teammates taking complimentary weapons. I would say maybe Retro Lancer and Gnasher for one person and Lancer and Sawed-off for the other player. When weapon drops become available (they don't appear at the start of the match), the Retro Lancer should probably then be dropped for the more powerful weapon. This will leave the Lancer to cover the rifle spectrum in most cases but allow the person with the power weapon to keep the Gnasher as for its versatility so he's not completely helpless at all but close range.
Of course, that also depends on the map and who you're playing with. All I can say is that it's usually a very fast paced game that is completely about team work and thinking on your feet. Make sure you keep moving because if you stand still, you're more likely to attract more than one team, putting you in a bad jam.