First off your battery should get a proper test. Until tested otherwise, consider it the weak link. (i.e. everything works, just not when you use YOUR battery.) So we don't really know the battery isn't culpable, just that the mechanic was able to sell you a starter.

You would get better mileage with some pictures/video (w/audio) preferably or expand on information.

Did it try to turn starter with battery only?

What is voltage on terminals when running?

You can get Factory Service Manual on Ebay usually for cheap on a thumb drive. But it's basically a combination of having the wiring diagram and following trouble shooting steps with a multimeter to isolate electrical branches.

TL;DR: Start at the beginning, don't assume something is good because something else was replaced. But if a jump starts your car, and your battery doesn't.... That narrows the field.