Dave:
Weather in June is usually pretty good. Sometimes a late snow can fall in early June. Late June is typically dry since the Monsoons haven't started yet.
The park is large, and trails abundant. One large cluster of trails eminates from the Bear Lake area. The most popular of these is the trail to Nymph, Dream and Emerald Lakes. These trails are nice in the early morning, but become crowded by mid-morning. In June, expect snow (maybe a lot this year?) in patches and ice in/on the higher lakes. Many trails from Bear Lake lead to high tarn lakes in glacial cirques such as Blacks lake.
I would recommend a trip to REI or some other store to pick up a book on trails in RMNP.
Glenn