Watch out for the altitude, its amazing how much of a difference it can make on energy and sickness. Do train pretty well, because depending on how you break it down, it can be very strenuous. If you do an RMI-esque climb (one night, summit day leave muir around 12-1am) it'll be hell, so break it up a bit more and enjoy it more.
Practice a shit ton of crevasse rescue skills. In the early and late spring, crevasses arent as much an issue, especially on the main routes (but check updates often), but they do happen. Quite a few people fall into them, but most people are fine with proper practice. If you aren't as tech, you can go up the main route (DC route) and usually RMI has fixed lines and shit, and even ladders if necessary. Lots of people go on that route, so its practically a trail up at times.
Weather in the spring can fluctuate far more than mid summer. I remember going up to Muir... was perfectly sunny leaving Paradise, but was practically a blizzard by the time I got to camp muir. So have alllll the shit you need.