Based on my experience down there if I were going the first week in June I would try the bayside creeks. There are not a lot of boat ramp options on the creeks themselves, but you could launch at Cape Charles or Morleys Wharf and run to Hungars, Plantation, or one of the other creeks and target specks (if this past winter didn't affect them too much?).
On the oceanside, I think that time of year is mostly for the big red drum in the inlets which i know very little about. I'll defer to others on that. Just gotta use good judgment fishing the inlets. As for running around down there, try to launch around low tide so you can see the oyster bars as you head out away from the launch and then you'll have your GPS track to follow at higher tides. At high tide you can run just about anywhere. Its the intermediate tides that can ruin the day. I don't take shortcuts during those times. To be honest the barrier island backwaters is a tough place to fish unless you put in a lot of time over the years. The bugs are the least of the problem. Its usually breezy (windy) enough that once you leave the ramp you'll wonder what the fuss was all about. The most annoying thing for me is the floating sea grass that the wind churns up and then drifts out with the tides. Deep, fast, turbid water is very typical. I use lines that will sink pretty fast.
Mid to late June through August brings tarpon and lots of sharks (just missed one on a fly last time down there), along with hordes of croaker anglers from NC. Even ran into some tripletail one summer. That is when I like to fish down there. Its a very cool place. Very humbling though.
Good luck with your trip. John