Try to play objective games like Demolition, Headquartes and Capture the Flag. Squeezing the EXP is crucial for levelling up, and also, always head for the objective. Hitting the killcap in Demolition is pretty useless, as you won't get more experience points than people who go for the objectives.
Also, complete challenges. Use a variety of guns and perks, and while you get used to them, you'll also complete various challenges along the way!