Leasing an apartment - I have to do this before I start training, right? (Regional BB, 1st yr analyst)

I have training in New York, but I will be working in a regional BB office (SF, Chicago, LA, etc.). Since training lasts for almost two months, I don't really need to move in to a new place until after training. But since I'd probably have to work really soon after training is over, I won't really get a chance to find an apartment after training, right?

Are there apartments that offer immediate start? I looked on Craigslist, most of them say "available from July 1st" - this means that those apartments require one to rent starting from July 1st (and not a later date), right? And usually first month's rent and another month's rent (as a security deposit) is due at signing?