Early June should be reasonably busy because school's out. That being said, the discrepancy between attendance on weekends vs. weekdays is a bit more pronounced in Busch Gardens because they rely more on "passholders", who tend to be local and who usually go on weekends. Essentially, you can buy a yearlong pass for less than the cost of two one-day admissions, so Bay area residents add to the crowds mostly on the weekends. By contrast, the Disney parks are often more crowded on weekdays than weekends, because people vacationing from up north and abroad tend to be en route on the weekend. So to minimize lines at Busch Gardens, I would suggest planning your visit for the weekday, and to arrive early in the day, if possible, when the coaster lines haven't built up yet.
If your kids are into coasters, Busch Gardens is the Florida theme park. The new Sheikra ride freaks me out just to look at it, with its 90 degree drop and inverted loops, etc. There's about half a dozen other worthwhile coasters, including Montu, which is my favorite because of the way your legs are left to dangle precariously. Yikes.
