I don't want to get big and muscular. Should I avoid free weight exercises?

Absolutely not. Many prospective weight trainees avoid free weights because they "don't want to get big and bulky". Free weights will help you strengthen the muscle, however excessive muscular size will primarily be determined by your diet and supplementation, in addition to training. If you're not eating to gain large amounts of muscle, then in this case, you'll have nothing to worry about.

If I'm training for strength and size, is it okay to only use machines, or should I use free weights as well?

When training for strength and size, to maximize your gains and to progress as quickly as possibly, you should absolutely utilize free weight exercises. In particular compound free weight exercises such as squats and deadlifts. These exercises work a wide range of muscles and will give you much better results than machines alone.