I'm missing my first June in years right now. Go for me, will ya?
THe crowds begin to pick up the second week and build from there, culminating in the July 4th weekend and staying peaked for about six weeks. In later June, you'll find heat and crowds. Managing them is a slight artform, but can be done and a great time can be had. The important thing is to come to the Disboards and get the info others just don't have!
To repeat EEyorelover22, use fastpass and EE. And as for E-Night, I personally think that's always a good expense. I'd personally do the following, depending on what you want to do:
Hit the EE park of the day, then get out by noon and head somewhere quieter. Avoid MK on Saturday at all costs. In fact, if you're allowing for an "off day," Saturday is the day to do it. If you're doing a water park, do it mid-week (Wednesday is a great day for waterparks). If you're doing DisneyQuest, try to do it on a weekday, and for heaven's sake do it when the sun is out. DisneyQuest on a rainy Saturday can be frightening.
This is one time to have a commando-style plan. Make good use of early morning, fast pass, and common sense. Example: in MK, run, don't walk, straight to Space Mountain, hit fast pass, then get over to Timekeeper. In less than an hour, you've managed two high-profile rides.
To beat the heat, and the crowds, look for large capacity dark rides and do them in the heat of the afternoon. Example: Hall of Presidents in MK; Ellen's Universe of Energy or O' Canada in Epcot, Who Wants to be a Millionaire in MGM (sorry, I don't do AK). Even if there's a line, they hold a large amount of people, they're extended (half an hour or so), and they have air conditioning.
Be willing to walk away from an attraction. We love Splash Mountain, but I'm not willing to wait in line an hour to do it. And I think of what I can do during that hour. Or, if you don't think like me (SIL would rather spend the whole day in line for SM rather than miss it), be mentally prepared. But I find I can have a more enjoyable day in the park and have more of a light-line feel to the experience if I go in knowing I may not be able to do SM, and if I instead do three other attractions during that hour.
If you want to see parades, see parades! But if you can live without them (or are in the park a second time and have already seen them), what a great time to hit an E-Ticket.
Go early. Leave mid-afternoon and take a swim/rest/etc. Come back around 4:30-5. Enjoy the line of hot, tired people who didn't take a break streaming out looking po'd. Then from 5-7, the standard dinner hour, enjoy a "lightening of the lines."
And Pat's golden rule: When everything else is busting at the seams, Epcot is just never too bad. It simply has a higher capacity than the other parks.
I've always enjoyed June. I also enjoy September/October for the short lines. The trade off is when Disney is more crowded, it just, well, looks more like Disney. And with the parks open later at night, you get the "lit up" factor going. Disney (especially MK), is just plain old neat in the dark.
Hope that helps.
Pat