While you can get the biggest amount of XP generally from King of the Hill, it also normally takes a little bit longer for those games to complete. It is, however, the best way for lesser skilled players, but this will be for the general Gears player.
Team Deathmatch has proved to be the winner here for me, OR I would recommend Wingman under the circumstances that 1) You have a partner that communicates and is good with ALL weapons, and 2) You can swallow your pride and use the sawed-off when necessary. Everyone will be different, so find the mode that suits you the best and get to killing.
General Tips:
1) This is not Gears of War 1 or 2, so don't go running around with your gnasher and always expect to walk away with a kill. Odds are you will die to a sawed-off regardless if there are level 1's or 40+'s in the room. Play smart.
2) Quick match games, as opposed to ranked games, seem to have far less sawed-off users for some reason. The XP ratio doesn't seem as high, but if you're a skilled player you can probably tear through opposing teams and gain a ton of experience that way.
3) Man up (down?) and use the sawed-off if you need to. It has an advantage in almost EVERY close-quarter situation. I know some people will say they've adapted and can kill sawed-off rushers, and I'm not disagreeing. I'm getting to that point... but the amount of players using it is so high that it may be your best options for easy kills/points.
4) Clicking the in the left stick while aiming at an enemy will spot them. You get assists for this, which earn you pretty much free experience. Putting a couple bullets in someone that gets killed by someone else will also net you assist points, so do these things.
5) Use executions when you have the opportunity. They give you bonus XP for using them. Just don't get greedy and do it mid-match, leaving yourself open to die.
I know these facts are somewhat basic, but for those looking at this from an achievement perspective, these are general tips that will also help you get better at the game.
*Edit - IDK why it is saying I don't have the achievement. I've had it almost a full day and have turned my Xbox off for the night and everything... So yeah!