Hi Kara,
I think on your family trip you will experience both extremes. During the Memorial Day weekend (especially the 30th and 31st) the park will be crowded with people taking advantage of the long weekend. However, by Monday morning the crowds should begin to thin out as everyone heads back to work.
The first week in June typically does have fairly low crowds. The holiday weekend has ended, and most schools are still in session during this week. I've traveled with my family at this time, and while the crowds were not as low as they are in January or September, they are quite manageable. I still recommend taking advantage of Extra Magic Hours if you are a resort guest and making any necessary dining arrangements ahead of time.
During less busy times, Walt Disney World tends to attract conventioneers. There is one popular convention that takes place during the first week in June throughout the Orlando area. Certain theme parks are on the agenda each day:
- June 4 - Animal Kingdom
- June 5 - Hollywood Studios
- June 6 - Magic Kingdom
- June 7 - EPCOT
You may choose to avoid these parks on the scheduled convention days to avoid the larger crowds that may be expected.
Have a magical vacation!