I'm considering purchasing the new ipad. I have zero experience in this area.
Sure you do! You walk into a store, tell the nice man or lady what you you like to buy, and hand over your credit or debit card.
How hard is this?
Does the apple store sell the new ipad with 4LTE?
Yes.
Do I have to go to either Verizon or ATT to purchase the new iPad?
No.
Do either of these carriers offer apple care at the time of purchase
Yes. Make sure though that you are buying AppleCare+ specifically, and not some insurance or other warranty plan that a cell phone store rep might try to sell you in its place. Otherwise It'll likely end up costing more, and you won't be dealing with Apple if you have a problem.
If you buy AppleCare+, you'll get a white box with a red Apple logo that looks something like:
If they refuse to give this to you, then pass. You still have 30 days after you buy the iPad to get AppleCare+ directly from Apple (though you'll need to take your iPad to the Apple Store to get looked over).
Honestly, you're better off just buying at an Apple Store from the start, as opposed to a carrier's store.
is apple care worth the extra $$?
It depends on how much you will be using it, and how prone you are to accidents. If nothing ever happens to your iPad as long as you own it, then obviously it wasn't worth it. However, it will pay for itself the very first time you crack the screen, or spill water on it, if it fails outside of the one year warranty.
AppleCare+ for the new iPad is $99, plus a $49 charge for replacing an iPad with accidental damage. So, $148. Compare that with $299 for standard out-of-warranty replacement without AppleCare+.