Calisthenics. There's nothing fancy about a calisthenics workout, but just because it's basic doesn't mean it's not effective. A 2017 study in Isokinetics and Exercise Science put a group of untrained individuals through a calisthenics program to test the effectiveness of an exercise method that uses no equipment. Participants did a brief workout consisting of four or five exercises three days a week for eight weeks. At the end of the trial, all participants had significant improvements in posture, strength and body composition.
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.