Some good advice above. As for the "where to camp" part of the question, coming out of Bullfrog we ask ourselves if a Rainbow Bridge trip is in the plans (maybe we have guests who have never been, etc). If so, then we head south. The decision making at that point is based on how close you want to stay to the marina. Seems like we always have someone who can't come the whole time - requiring a speedboat trip to the marina at some point. The Rincon is about as far south as I go if a return trip is required - and preferably not even that far. If there's no return trip required, you've got lots of options between Bullfrog and Rainbow Bridge - including the awesome Escalante and San Juan arms. Plan to do a little exploration with one of the speedboats into the various canyons to find the right spot. You won't be able to easily turn that floatilla around in a narrow canyon.
If Rainbow Bridge is not a concern, you'll find more seclusion heading north. In June you might get lucky with one of the nearby canyons (Moki, Hansen, Crystal, Forgotten). If not, once you get past Forgotten Canyon, the crowds thin out quickly.
As always, lake level is everything in determining beach availability.
Finally, if you're okay with neighbors and full sun, Halls Creek Bay is close by and somewhat similar to the open shorelines found on the south end of the lake. It can be a great base of operations for exploring both uplake to the north (Defiance House ruins, etc) and downlake to the south (Hole in the Rock, Rainbow Bridge, etc). It's right next to Lost Eden Canyon which is a fun spot for swimming in the shade when you need a break from the sun.
Hope you have a great trip!