We are both new to MTG. He will attack me, and when I decide to take the hit and roll back my life, he then likes to attack me a second time. This doesn't feel right to me, is this allowed? Can he hit me twice after I've blocked or taken the hits?
No, you can only attack once per turn
There are certain cards that give you a second attack, but normally, one attack per turn.
If you encounter situations not quite covered there, you can always check back here, or try to figure it out yourself in the Comprehensive Rulebook (don't use that one to start out).
No, this is not correct. You usually only have one combat phase (There are exceptions using cards that give additional combat phases.)
Imagine this is not the case and you don't have any blockers. Then you could be dead on turn 2 by a 1/1 creature attacking as many times as you like.
As most have pointed out, there is only 1 attack phase.
The exception would be if a card explicitly states a second combat phase, e.g. Aurelia, the Warleader:
Whenever Aurelia, the Warleader attacks for the first time each turn, untap all creatures you control. After this phase, there is an additional combat phase.