Weight training will help, but remember, bulk is unnecessary and may actually be counterproductive to your goals.
Doing one day per week of various dumbbell exercises and some other free weights will be good.
As mentioned above, work hard training the actual events. You get better at push-ups by doing them, many reps of them regularly. Same with running and sit-ups.
You can do other body weight exercises as well to round out your workout.
Examples may include: jack knives. Leg raises, Russian body twist(with or without weight), weighted squats, get off your back, different types of push-ups, etc..
When running, include running hills, and run on days when the weather sucks as well as nice days.
Finally, be sure to hydrate well, eat well, and get proper rest and recovery in between training days.
Consistency is key, remembering that this will be your lifestyle when joining the Military.