You start getting paid during your Initial Entry Training. Signing bonuses are tricky... those huge 20k bonuses you hear about are broken up into installments. You'll get your first half after you complete initial entry training, and then they'll break the rest up into installments and you'll receive the rest throughout the course of your first term of service.
Your first 'paycheck' will be a direct deposit into your account about 2 weeks from your ship date to basic training. The initial portion of your signing bonus will be paid shortly after you report to your first duty station (after basic/AIT). Be sure to take your bank information with you to basic to make sure it gets in on time. Otherwise you will have to open an account there with a local bank. Remember that your bonuses are taxed like regular pay (in the military your pay/ re-enlistment bonuses are not taxed if you are deployed to a hostile fire zone).