It's crucial to stretch and work on range of motion before and after each workout as part of your warmup and cool down. Before your workout, warm up for five to 10 minutes with light cardio; then do some dynamic stretches and mobility exercises. If you're going to be doing a leg heavy workout, for example, pay special attention to the joints and muscles in your lower body.
Many of the activities you do to build strength will also help your balance. Balance is partly a matter of developing your small stabilizer muscles that provide support and keep you steady. Free weight exercises, such as lunges and deadlifts, will help strengthen the stabilizer muscles. Even better, include some single-leg exercises, such as single-leg deadlifts and pistol squats, in your strength-training routine.