I have been lent a car and I took it out for the first time last night (it's a 2003 Chrysler). The car had been left for a while and the battery had been disconnected. I connected the battery and turned the key. It turned over well, ignited and then died (my foot was on the accelerator and it got up to 4000 rpm before I took my foot off and it died). I repeated the same thing 3 or 4 times. Then the next time, nothing happened when I tried to start the car (i.e. no starter motor noise, nothing but clicking).
I disconnected and reconnected the battery and then the car started the first time (no idea why or what difference this could have made); drove it for 30 mins; (it was dark, but the lights were working ok; the oil light was on for a while, but went out after 15-20 mins); parked for 15 mins; started without a problem; drove for 15 more mins; parked for 2 mins; started the car fine; drove for 5 mins then parked. When I tried to start again (30 mins later?), I had the same problem as initially: igniting then dying 3 or 4 times; followed by nothing happening except clicks when I turned the key.
I called AAA. They came (checked battery voltage?) and said since nothing was happening when I turned the key, it was probably a problem with the starter (I may not have fully described all of the history because I hadn't collected my thoughts at that point). When they gave me a boost, the car started fine.
My current plan is to take the battery to a shop, where they will check voltage, cold cranking amperage and do a load test.
So my question:
What's likely wrong here? Battery, starter, alternator, or other?
Thanks very much for any hints!