You can try Whitney- but getting a permit in advance is going to be an issue- you've missed the lottery window.
I wouldn't recommend cutting your teeth on Shasta or Rainier. If you don't know how to properly use an ice axe and crampons, your life is potentially at risk. Better to try them out in Lassen NP Park somewhere, before a trip to Shasta. Better yet you could book a 1/2-1 day course with a guide service in Mt Shasta City. If you feel comfortable after that and talking to the instructor, you'll be in good position to climb Shasta the next day. And if you don't feel comfortable, there are MANY things to do in the area so your trip isn't wasted.
Definitely don't climb Rainier until you are familiar with glacier travel. Need to know how to use a rope, crevasse rescue, etc... as does your partner. You can book a guide service for Rainier, but I wouldn't recommend it. From the looks of people who did, it seemed like a pretty miserable experience. Getting drug up a mountain like that, following their timeline, by a company only wanting to have a high client turnover ratio... well... there's better things to do with your vacation.