the voltage should be around 12v or slightly higher when there is no current draw. when the car is running the voltage should spike due to the charging of the alternator. A new battery shouldn't die or drain that rapidly if there is a charging issue. You may just have a poorly charged battery or bad one. Put the battery on a charger and trickle charge the battery overnight. the lower the amperage and longer the charge the better. since it is not completely dead you won't need to use high amp. charging. after that, install it again and see what your results are. Then if you still have problems move to the charging system. Test at the alternator for proper voltage. check engine light usually flickers or comes on when the alt is going or gone. wouldn't hurt as suggested to check the ground strap from the body to the oil pan on the left had side engine bay. This is the main ground.
Lastly, make sure your battery cables are securely attached to the terminals.