Q: car won't start with new battery

I left my lights on a week ago and the battery died. Had triple A come out to jump it, and it didn't want to start. This has happened before, whenever I've had to have it jumped it's been hard to start, but after the jump there has never been an issue with getting it started. It clicked the first day afterwards, but I let it sit for a week because I didn't have any time off to get it looked at. Thought it might have been the battery even though triple A said the battery was fine and had a full charge. Had it tested at a local auto part store and they showed me it had no charge. Bought a new battery (old one that came with the car was a triple A battery that was dated for 2012), installed it correctly (positive first, negative second, nothing turned on to blow a fuse), and now it won't even attempt to turn over and nothing turns on. Is it possible it's the starter or did I maybe get a bad battery? Or could it be something else?
My car has an automatic transmission.