Q: How do I tell the difference between an alternator issue and a starter issue?

If the starter motor is turning the engine over at adequate RPM, then you definitely do not have a starter motor issue. Even if you had a bad alternator (and the alternator should absolutely be tested, certainly when you install a new battery because you want to be sure the charging function is working properly) that would not prevent the car from starting. So, if the starter is turning the engine over, but the motor won’t run, you have either a fault in the fuel or ignition systems, or the air induction system. To "quickly" determine if it is a fuel system problem, you can try introducing starter fluid into the intake while cranking and if the vehicle starts at that point that would be an indication that you have, at least, inadequate fuel delivery perhaps due to a fuel pump issue. The most efficient approach to getting this resolved is to just simply request a no start diagnostic and the responding certified mechanic will make quick work of this for you. At the conclusion of the diagnostic he will let you know of the fault and the cost to repair. Please let us know how we can best assist you to get your vehicle running properly.