01 Jeep Grand Cherokee starts running and then stops after 2 seconds

That is usually a sign that the ECM did not recognize the key. Thinks it is being stolen. Shuts down as theft protection. Should also flash an icon of a key with a line through it. Also check your keychain to make sure other car keys aren't close enough to cause a confused signal. Could also be a fuel problem. Check pressure on fuel rail...about 55 psi.

So what do I need to do to fix it? Go to a dealer get a new key? We just bought the car yesterday... So frustrating!

And yes I'm quite certain that it is the security system blocking the key. There is an icon of a key flashing as well

I'm sorry your new Jeep is getting off to a bad start. Maybe unhooking the battery for 10 minutes may reset the ECM. It might just be a glitch. If you drove it home, it has to be the right key. Some states allow you to reverse a sale within the first 36 hours. Buyers Remorse Law. Or stop payment/loan/check. That will get their attention. Most Jeeps come with two keys. Try the other one. Lenders often hold back one key in case they have to repo for non payment. Find that other key. I hope the best for you.