First, your current alternator is just fine, no need for a larger one.
If you installed a new battery, and the starter doesn't turn the engine over, a couple things: dirty/corroded battery cable connections to the battery, fender relay, or starter.
The cables themselves can be bad. Look for corrosion under the insulation at the battery terminal connectors. Make sure the battery cable to the starter is in good shape.
If the terminals are clean and tight, and the cables are solid, then you may have a bad starter. You can try jumping the large fender relay terminals with a jumper cable. Either use one side of the cables, or connect the two ends of the cable to make one wire, then jump the terminals. This feeds voltage directly to the starter, and bypasses the relay. If the starter doesn't turn, bad starter.
If the starter now turns the engine, the fender relay may be bad. There is a small wire going to the relay. This wire comes from the ignition switch. You can use a wire to jump from the positive side of the battery to the small terminal. If the starter now spins, you have a problem with the ignition switch or the wire going to the switch. If it doesn't spin, possible relay problem.
All the alternator does is keep the battery charged, it does not affect engine start. If the alternator is working, there will be over 12.8 volts at the battery with the engine running, usually 14 to 14.5 volts, but not over 15volts.

shady