Sounds like a bad starter to me.
I'd take it to an AP store and get it tested (ask them to do it repeatedly, so it gets hot).
Other than that, I'd say bad cable or battery terminal, so give them a good inspection. A lot of battery cables corrode underneath the insulation right at the battery terminal end, so it is hard to see. If it tests good, I'd replace the red cable to it.
Does not sound like the selenoid.