If you take the pen and paper ASVAB, it usually take around a week or so to get your scores back. If you take the computerized version then you will receive them immediately.
Now, when at MEPS you will carry all your paper work the doctors have to fill out with you. If you are like me, you will be checking the results after every individual part of the physical. Once you pass, the head doctor will give you your paper work and tell you to take it to your liaison. If for some reason you don't pass however, they will let you know for what and why. Here is when my recruiter told me to fill out a permission slip to check my credit and dismissed me. After MEPS, they will wait for your credit check to comeback along with your blood test and urinalysis. If everything is spiffy they will then have you do your clearance paper work. Some recruiters will also have you DEP it at this time and try and get you a ship date.
These are my steps, and every MEPS/recruiter is different so don't expect this to be the text book answer. I have read of people waiting upwards to a year or two after MEPS to get a spot, so I hope you have patience.