The beginning of June is a wonderful time to visit Walt Disney World, as the weather is beautiful and the crowd level is good, too.
According to touringplans.com, an unofficial crowd-predicting site, the parks will be anywhere from 1-6 on a scale of 1-10, with the majority of days toward the lower end of that range. Many schools across the country are still not out for the summer yet, so you'll have locals and families with preschool aged children or no children at all traveling during that time of year.
The weather is typically 91 as a high and 71 as a low, according to Weather.com, so your days will be warm enough to enjoy water attractions, water parks and resort pools, and evenings will be pleasant as well.
Send more questions as you think of them, Janet; we love to help!