02 monte carlo

Have an 02 monte Carlo and needing some help.
Got given this car and was told the starter was out
and needing a battery cause there wasn't one in it.
Ok got new battery got new starter well it fires right
up start going down the rd and battery light comes
on so we turn right around and go back home. Go
have the alternator tested at a shop and its bad the
brushes went out get that fixed go home put it
back on. Go to have the battery recharged so the
alternator isn't working aonhard to charge the
battery back up. Get back home put the battery
back in go to hook it up and then it wants to try and
fire up with the keys still in my
pocket....ummmmm???? Any ideas.