Good suggestions from folks above. A lot depends on which way you are coming in, and your willingness to go up and over Trail Ridge Rd, with whatever you're driving/towing.
On the west side, the "quiet" side, there's the Winding River Resort that was mentioned, with amenities, and within RMNP itself is the Timber Creek CG; dry camping in an area that was devastated by the beetle kill, with many trees cut down, and only stumps left in the CG. Around Lake Grandby, south of Grand Lake, there's 4 USFS CG around the lake, notably Arapaho Bay on the south end.
On the east side, in and around Estes Park, there's been some good suggestions; within the park, there's Morraine and Aspenglen CGs. If you're willing to travel a bit, south of town along CO7 and 72, there's Peaceful Valley, Olive Ridge, Camp Dick, and even further south, Brainerd Lake. These are all USFS and dry camping; they fill up very quick during the summer, so if you're interested, look into reservations now.
Estes Park at the East Portal is nice, but it's tight, depending on what you have. The back end is great for popups and the like, not so much for trailers or MH; these are best along the front end of the park. There's also Estes Park at Mary's Lake, which is much easier to get in to, and have spots ranging from full hookups to electric only, and a separate tent camping area. It's a County owned park (as is the East Portal) and the rates are more reasonable than the commercial parks.
For commercial parks, there's Paradise on the River, Spruce Lake, KOA, and Elk Meadows in town, and just outside is Yogi Bear's Jellystone Park. They are all going to be more expensive, and they also can fill up fast on the weekends.
Hope you enjoy your stay here!