Google search TC 3-22.20 (This is the Army's new manual for physical fitness. Download it.)
Yes, weight lifting (also known as strength training) is a critical part of overal physical fitness and will help you to perform better at all three events.
Alternate days of which muscle groups you work out. Do upper body on opposite days from lower body. You may want to do biceps and triceps on Monday, squats and lower body strength training Tuesday, back (lats) and shoulders on Wednesday, etc. Cardio and abdominal work are critical to do EVERY day. Vary the cardio workout you do, and ALWAYS do the weight training before you do the cardio because this will enable your muscles to absorb more oxygenated blood during the strength training before it moves back to the core during cardio (reduces risk of muscle injury and passing out).
Of course, you will want to engage in the activities you will be measured on. However, vary the type of push-ups you do. Switch from wide-arm to close-hand, for example, to work different muscle groups in your chest, triceps, biceps, and shoulders. Do sit-ups with a medicine ball or on an incline so you have greater resistance. Run short distances very fast, do pole-to-pole sprints, and then also do a long, slow run (like 4 miles or more). Keep it changed up.
Also, allow plenty of rest and recovery time for each of your muscle groups. Swimming is a great no-impact cardiovascular workout and can also provide resistance for all muscle groups.
Good luck!