How to tell if there is a probem with the.

Was the battery load tested? Use a digital multimeter across battery posts, everything off, what is the reading? With leads still hooked up, have helper turn key to crank. How far does the reading drop?

Just to add, at the starter motor, automatic tranny, red wire is hot all the time. Yellow wire with lite blue tracer goes hot when the starter relay is energized. Any testing at the starter, make sure tranny is in park or neutral and the parking brake is set.