Bought a new truck about a month ago, have had my old ranger parked since then. Went to start it up a few days ago and had power but no crank. Got the battery checked and it was bad so replaced it, didn’t fix the issue.

I just replaced the starter, but when I went to reconnect the battery, it immediately started to try to turn over on its own, without the key in the ignition. Not sure if this is a wiring issue or what. I have the signal wire on the 10mm stud and main power on the 13mm and I’m fairly sure that’s right but never swapped a starter before

 

Think that new starter has a short?

Pull the signal wire off (protect it from grounding on anything else) and measure voltage with battery reconnected but key in off position. Should be 0 voltage. Have someone cycle key to start and it should have voltage. Than everything is good. Get another starter as the attached solenoid may be stuck or shorted.

Here is a wiring diagram:

 

Welcome to the forum

You have a 2004 Ranger with 4.0l, should include that with questions, just FYI

Yes, it reads like you hooked up new starter motor correctly
Larger cable to top post on the starter solenoid, that's the smaller cylinder that sits on top of the motor

Then the smaller wire to the "S" post, yes it has a "S" next to it

As seen here: https://troubleshootmyvehicle.com/im.../image_T_1.jpg

When the "S" post gets 12volts it pulls the solenoid closed and connects the two larger posts together, so starter motor now has 12v directly from battery, and starts to spin

You have to make sure the Battery positive cable can't touch either the other larger post or the "S" post after installation

Also in the engine fuse box is the "starter relay", pull out that relay and connect battery cable, if starter motor still spins then its most likely a short in new starter solenoid, so take it back
If it doesn't spin then could be bad starter relay, but long shot on that