"You've got exactly five minutes to #$%@!, shower and shave, sweetheart, so pick your favorite two, and one of them better be shaving or I will have the rest of these undisciplined ladies dry-shave you while you doubletime on my quarterdeck!"
You're really not going to shave with a DE in basic, I hate to say. Even if you bring enough blades to last you a lifetime, they're going to get taken away in shakedown faster than you can say "safety hazard." They're certainly not available for purchase at the PX when you get the chance to go. I don't know what the situation would be for Trac II cartridges, which would probably be the best alternative. I guess you could have your family send you some, but there would be a decent delay. Whether or not you want to bring a brush and decent cream is up to you, but I don't think I would; being That Shaving Guy is not to your benefit. The best advice I can offer is to pack a Mach 3, embrace the suck, and save the great shaves for AIT or beyond.
As far as time, you'll have to wake up before reveille to do things anyway. I and most other people shaved then. If you're a night shaver, we always had around an hour of "personal" time at night for showers and a billion other things that needed to be taken care of, so you could do it then. But don't realistically expect to have any more than ten minutes to budget for a shave. I rarely took more than about three.
In basic training the best thing to do is conform and fly under the radar. Unfortunately this means you need a Mach 3 or other recent P&G or Schick product with multiple blades. You will not have the time to enjoy a decent wet shave while in basic, I'm sorry. It should go quickly though, they can't control time. That's one thing I vividly remember a sergeant telling us at reception before we went to training.
"you've got exactly five minutes to #$%@!, shower and shave, sweetheart, so pick your favorite two, and one of them better be shaving or i will have the rest of these undisciplined ladies dry-shave you while you doubletime on my quarterdeck!"