Straight Talk allows you to use your own phone on AT&T, TMobile and Verizon towers (but not LTE phones on Verizon towers, unless they've changed that). The way you decide which phone to get is by determining which of the carriers has the best coverage where you need coverage. Then get a phone for that carrier and get a Straight Talk SIM (micro, in the case of a Note 3) for that carrier. You can use either AT&T or TMobile Note 3s on the other one's towers, but your high speed data will suffer - how much depends on where you are. Since the cost is going to be about the same, buy the phone for the towers you want to use.
(They also use Sprint, but the last time I looked, they only allow you to use their Sprint phones.)
Don't depend on the carriers' maps - they're as accurate as a good guess. Talk to people you know who are on TMobile and on AT&T and see what kind of coverage they get - especially if there are any bad spots where you need coverage. (I'm not ruling out Verizon, but a) make sure you can use an LTE phone on ST on Verizon if you choose them and b) personally, when the signal "goes digital", I can't stand a bad CDMA signal - it sounds like the person is under water. GSM is closer to a noisy signal, and that doesn't bother me. (I also like the fact that I can switch phones by just switching the SIM card, and AT&T is fine with it - I'm not doing something I can "get caught" doing.) Switching phones on Verizon is more complicated, and if you're on ST ... I don't want to think about having to deal with their support people.
When I moved down here from up north, I had great coverage with Sprint on ST, except for one place - my daughter's house. Since we spend a lot of time there, it was time to change carriers. With my current carrier, there's a large part of town that has poor coverage - a part I'm never in, so I don't care.
That's how you choose your carrier.