We plan to go on Monday, Tuesday. Right now I am wondering if after this travel if sleeping in would be better and doing a half day Monday with a full day Tuesday? Then maybe Wednesday morning. We head for Hong Kong afterwards as we take on all the Asia parks.
If it's Monday, Tuesday, and Wednesday, before Mid-June I'd say do the full day Tuesday, and see if you need need Wednesday morning. After Mid-June, schools are on holiday so it's hard to say what crowds could be like, but if you get into the park earlier either through the hotel or through early access, you might still beat the crowds, but it's probably a good idea to go Monday if you're worried about "missing" something for crowds.
On Monday, you can hang out in Disney town, shop and relax, and maybe check out Wishing Star Park (I haven't yet.), and enjoy the hotel. If characters are a priority, try to check out BOTH hotel lobbies, there maybe an opportunity with less lines than in a park! If you decide to go to the park, on Monday consider watching the parade and fireworks that day, so that you can maximize your time on Tuesday and take advantage of shorter lines during those times, but also keep in mind it can be hard to get around the hub area during those times if you are running from ride to ride. Go to and from Tomorrowland through Toy Story, for example or you'll get stuck. Keep to the back as much as you can.
On Tuesday, prioritize carefully, but Soarin' 7DMT and Tron should be near the top, if you're interested at all and in that order. Book one as a fastpass from your phone AS SOON as you walk into the park (set the app up in advance and connect the party as necessary), and head to another. Hopefully, your fastpast is early enough, and you'll be able to book something else quickly. If you have early entry, check out smaller rides now, like Peter Pan or Many Adventures of Winnie the Pooh. While Pirates is a must-do, the capacity is SO HIGH, it often has the shortest wait, so don't prioritize it, but don't forget about it either. For shows, the performance windows end pretty early when I've been there, so make sure to check out the times for what you must see.