Does it turn over like it has a heatlhy battery in it but won't start? Or it turns over like the battery is dying or very low?

It is not starter motor my friend, or else it wouldnt turn at all, so eliminate that.

Here is what I think:

1-If it turns over like it has a dying / low battery, then it is most likely your alternator because as you said the battery is brand new. (Make sure to check connections between alternator and battery to make sure they are properly connected, before getting a new alternator.

2- If it turns over just like any other healthy car but it does not start, then your problem is not batter or alternator. It is something else...