only, I heartily recommend Alicia's Bed and Breakfast.
It is NOT on the beach, so if you want beach front don't bother. BUT the ocean/snorkeling/beach/etc is only a 10 minute cab ride away and cabs are cheap.
We stayed with Alicia and Chuco about 4 years ago; three of us. $35 per night with full breakfast every day. Small, impeccably clean, delightful hosts, camraderie with the other guests (max capacity is abou 13). Our own bath, AC, dog to pet, and plenty of good local advice about stuff to do.
There are several nice Bed/Breakfast places around the town.
Now there are many nice beachfront hotels, which are part of packages offered by various tour companies. Prices range ..we were going to stay at CasaDelMar until we found the aironly and went with the B&B instead.
It is just south of town, across the street from the beach and its sister hotel is on the beach. There is a tunnel under the road so you can go over to the other hotel where you have use of their facilities.
While you are there, you must go snorkeling. There are a zillion dive outfits, all are good. We went out with Dive Palancar and had a fabulous trip which departs from the Allegro.. We also snorkeled for free at Xul-ha, went out with an individual guide and also spent a day at Chakkanab, a local beach park which is very nice.
Also, get thee to the east beaches:
Plenty of good food, busy town square, shopping. Stay away from downtown when the cruise ships come in, and that is posted all over the place downtown. The heavy cruise days are the best days to go to the east side of the island. Native items and crafts are cheaper as you go deeper into town and real cheap on the east beach.
There are some neat ruins in the center of the island. We went to the ruins, then on to the east beach in the same day. Wonderful. Just be sure to take some bottles of water with you on the tour of the ruins, it is HOT.
You can ferry over to the Mainland to Playa del Carmen, Tulum (more ruins), but I warn you, Cozumel will lure you in and make you want to stay.