Last July, I visited the Cliffs and, separately, one of the Aran Islands, without organized tours, and without any crowds except for immediately around the Cliffs' visitor center. Both places are beautiful. There are some tours that cover both in one day but that might be rushed.
The main point I'd like to share is that the Cliffs of Moher is more than the visitor center. The cliffwalk offers miles of wonderful views of the water below on one side of the path and farmland and/or Burren landscape on the the other, for about 5 miles north of the visitor center to Doolin and about 3 miles south to Hagshead. Bus Eireann stops near where the trail starts in Doolin and again at the visitor center. A local shuttle bus also stops near the Doolin point, at the visitor center, and at the Hagshead parking lot on the other end. You can walk one segment and then catch a bus, or just walk a while and head back. The only part I saw crowds was for about maybe 1/4 mile closest to the visitor center.