I'm studying to be an athletic trainer and one of the thing that's always reiterated in my studies is "train for the activity the athlete will be performing".
Basic training is going to be a nearly non-stop workout. They're conditioning you, not only to be a military drone, but to have muscular and aerobic endurance.
So for you, what I'd recommend is training for this situation - and the best way I can think of is long sets of high intensity intervals with weight-bearing exercises interspersed. Plyometrics will be both your best friend and your worst enemy. Here's what I'd prescribe as a regular workout:
Warmup: Run (!) for 10 minutes
Do 5 sets of this with about 5 minutes of rest between each set:
1. Do sprints back and forth about 50 feet apart - at each end of the sprint, get down and do 10 pushups. Do about 10 full laps of this.
2. Do sprints back and forth about 50 feet apart, except at each end point, and right in the middle, do a pushup, pull your legs inward toward your body, and then do as high a jump as you can - then go to the next point.
3. Grab some freeweights, hold them to your side, and do hops (from a squat position, butt low-to-the-ground as possible, hop forward as far as you can without injuring yourself). Do this about 30 times without rest.
4. Do sprints back and forth for about 50 yards, and touch the ground and the end of each sprint. Do this until you really, seriously can't anymore.
I've found workouts like this are nearly impossible to do alone, so if you can find someone to coach you (and on alternate days, they can do the same workout and you can coach them) that would be best. This is the kind of workout my trainer (who was a Marine Corporal) had me doing for a couple months, and my muscular endurance skyrocketed (I went from being able to do 15 pushups at a time, to being able to do 60 - 70 pushups at a time).
