The earlier you go in June, the better. May 31 is not Memorial Day weekend, so you shouldn't have to worry about those crowds. As it gets closer to the end of school, more and more schools will start having field trips there, so you'll see a lot more crowds of 10-18 year olds as it moves into the second and third weeks of June. The dates you mentioned should still be ok.
A good way to know when it's considered less crowded is to see when the season passes are valid. The least expensive passes (So Cal) have a lot of blackout dates - if those passes are blacked out, expect more crowds.
If you get the chance to go again, early March (before spring break), late April (after spring break) and early May are all good days. September and early October can be great too. Surprisingly, days like Labor Day and other Monday holidays are often not that crowded at all (though I would imagine the Saturday & Sunday of those weekends are).
As for lines, within Fantasyland, Peter Pan & Dumbo will have the longest lines, along with Alice in Wonderland. Do these first thing in the morning or during lunch hour for less crowds. Star Tours is a long wait - get a fast pass and bypass the crowds. Small World usually has a short wait. Go in the outer (right) line as opposed to the inner (left) and you'll move a lot faster. Autopia is often a 10-15 minute wait, but also can easily be a 30 minute wait, so go there if you notice a shorter wait since it won't last.
No need to do both Casey Jr train and the Storybook Boats, as they take you to basically the same places and you see the same stuff. Personally, I could skip both, but if the kids want to do them, choose one and not both.
Teacups almost never have a line at all.
Are you also going to CA Adventure? If so, get there before it opens and head immediately to the new Cars Land ride. I forget which it is, but one of the rides is by far the most popular and will have a very long wait. Toy Story Mania is also a great ride that always has a super long line if you don't do it first thing in the morning. Not sure if the Toy Story lines have shortened now that Cars is open. Also not sure if either has a fast pass; if they do, utilize them.