I woudn't open up any more than two cards at once. And I think it would make sense to have card #1 on one's report before applying for card #2.
I'm a big fan of one card every six months, i.e. card in months 1, 7, 13, 19, and 25. This keeps you off the radar for two many inquiries or new accounts. It always keeps you below 5/24. And it positions you for a Chase card in month 13. I think it's OK to deviate from the schedule a bit as long as card #5 doesn't come sooner than month 25.