The largest concentration of Ancestral Puebloan sites is at Chapin Mesa, where you'll see the densely clustered Far View Site and the large Spruce Tree House, the most accessible of sites, with a paved half-mile round-trip path. Spruce Tree House is currently closed to visitors, but you can see it easily from the museum overlook.
If you want to see Cliff Palace or Balcony House, the only way is through an hour-long ranger-led tour booked in advance at the visitor center. These tours are extremely popular; go early in the morning or a day in advance to book. Balcony House requires climbing a 32ft and 60ft ladder – those with medical problems should skip it.