Gotta agree with Tar Heel. I go every June, usually the first week, sometimes the second week. I also try to go in September or October. The crowds, while a hefty increase from the value season, are managable, even when compared to the third and forth week of June, when the summer-break families start piling on. We've found the rain, like most other times we go, to be a hit-or-miss shower most every days, and the chance of one extended storm while we're there. All manageable. I think late September is the time to go, but I've always enjoyed my June visits. Proper park management, meaning using FP, taking a break mid afternoon (and coming back at night), and getting the long-line rides in early, all yield a pretty nice trip. We normally see no wait more than 20 minutes.
As for Gay Days, I'll also agree with Tar Heel. I've seen this make absolutely no difference in our trip, but due to massive, massive crowds, I'd also avoid MK on Saturday. And for the record, I'd avoid PI on Friday and Saturday night. I'd also find out the days events are held at TL (two of them, I think) and avoid those due to volume.
Enjoy June. I hope I'm right, 'cause I'll be there with ya!
Pat