You really need to do both. Cardio will help burn fat, but weightlifting is also important in order to build muscle. The more muscle you have, the more there is to actually burn up calories.
Cardio is also important in order to keep up you VO2 Max and stamina (good for paintball!)
Weightlifting will ensure that you wont tire out as fast, (also good for paintball cuz you can hold your gun up longer!)
It is possible to do both at the same time. High intensity work outs that use a lot of push ups, sit ups, pull ups, squats, thrusters, plank, whatever all done one after the next (ie, one set of push ups, move strait into sit ups, strait into squats...no rest in between...then back to push ups...) will keep your heart rate way up and act as cardio at the same time. As a bonus they all work your core muscles, which will inturn provide a solid base for your outer muscles and when the fat begins to trim away you will be left with a hot 6 pack!
I also throw in skipping after every few sets to keep the heart rate up, it is supprising how much of a work out skipping is. There is a reason boxers do it!
My business card says "You're in luck, I do two things..."