If you hike Whitney anytime before the end of June you're going to be going hiking through a good bit of snow. It's been a big snow year in the Sierra this winter.
You have to get a permit to hike Whitney and virtually all of those are gone from the summer since the lottery for those was in February. But if you drive to Lone Pine and go by the ranger's office, you might be able to pick up an unused permit from someone who is a no-show. (That happens almost every day.)
Read up on Mt Whitney and all the web pages from the Inyo National Forest about hiking it. Also read about the effects of altitude if you haven't hiked about 10,000' before. Whitney is a very simple straightforward hike for a 14er, but still people do stupid things and get hurt.
Shasta is a whole other ballgame. Gotta have crampons, ice-axe, and a helmet (falling scree). Shasta has only about a 40% success rate (even by experienced climbers) because the weather can change so quickly since it's further north and isolated. If you've never done any hiking on snow and ice, it'd be wise to guy with a guide service up Shasta. Read the online accident reports for Mt Shasta and you'll realize how easy it is to die on that mtn. Also, beware of those fumaroles.