Battery to starter, As big of cable as you can afford, welding cable is flexible and works just as good. 1/0 to 00, 4 gauge is not big enough.
Starter to coil, 18 gauge minimum, 14-16 gauge if you have it. 10 gauge not needed. Alligator clips will get the job done, not a lot amps here.
Alternator or generator, 10 gauge minimum, 4 gauge if running a lot of accessories, 8 gauge is probably about right. No real need to wire up the alternator unless you plan on a lot of run time.
I'm on the side of, you can never have to big of a wire. But with that comes routing and connectors, the bigger the connector the more they cost.
I strip the plastic off crimp connectors, add shrink sleeve to the wire and solder all my connectors on, then I shrink sleeve over them. To me they look more professional.
I made my whole wire harness from a harness I stripped out of a school bus. It had all numbered and colored wires 10 to 16 gauge wires. Its big and heavy duty, but in the last 26 years I have never had a light flicker or the truck never start.
On my engine test stand I built for V8 engines, it has 000 battery cable coming off a 950 cca battery. It has never failed to start a new engine. I use alligator clips on the wire ends for easy hook up and mechanical oil and water gauges.
How to crimp wires,
https://www.jalopyjournal.com/forum/threads/crimping-tutorial.1127519/
