On the positive terminal, there should be a big fuse. Try replacing it. It's not expensive. Check all your fuses and relays. Disconnect the positive and negative battery terminals and clean every single connection. If the cables have any green showing on the wires, just cut and replace them correctly. Check the wire to the fuse panel under the hood. Check under the fuse panel. Check the grounds. The alternator comes in two variants - automatic transmission and standard. Make sure it's the right one. I know it's pricy, but call the dealership, give your vin and be sure your parts in question are correct to your vin.
These are the absolute easiest cars to work on. Very basic bolt on parts.
If your car happens to cut while cruising and the battery light flickered while feeling a brief cut (doesn't die, but cuts and restores at random) then you should check your crankshaft position sensor, and cam sensors. They are cheap from the dealership. Stick with OEM. Do not buy a cheap alternator. If you are on a budget, go on eBay and buy an OEM used alternator from a high rated seller.
