Welcome aboard.
The Disney parks stay open later than the Universal parks. Looking at the June hours for the MK on June 14th, they stay open until 11pm, while USF closes at 9pm, and IOA closes at 8pm.
It's better to visit Disney first and get used to the long lines. On the day you leave, visit AK, because that park only takes half a day. Show up early and ride Everest first thing.