To clarify the above post, Woodlands on Fall River is inside the town limits of Estes Park. It is located along the Fall River Road corridor as are many nice lodging establishments. Lodging along Fall River Road is close to the Fall River Entrance to RMNP. In addition to the Woodlands, Streamside on Fall River, Estes Park Condos, Riverstone, Stonebrook, Riverwood, Aspen Winds, Pine Haven--are just some of the many options. There are not that many lodging places actually located in the downtown area of Estes Park. Elkhorn Avenue is the street going through downtown, and as lahord59 stated, it can get quite busy.
I failed to mention that in addition to the cvb site lodging info, people have had very good experiences with www.vrbo.com when looking for lodging in Estes Park.
Lodging close to the Beaver Meadows Entrance to RMNP would generally be along highway 36 or highway 66. Some in those areas include Alpine Trail Ridge Inn, Rams Horn Village, Colorado Cottages, YMCA of the Rockies (a resort not your typical Y), Swiftcurrent Lodge, Valhalla Resort, Rockmount Cottages, Glacier Lodge and Paradise on the River.
In a "normal" year, you should not have to worry about snow inhibiting most trails by late June. Of course, 2011 was a rare 300% snowpack year and this was not the case. In contrast, 2012 was a 30% snowpack year and trails were clear very early. Rangers at the RMNP visitor centers can give you up to date trail conditions and suggest hikes as well.