Verizon is a CDMA network. AT&T and TMobile are GSM networks (actually TDMA at that level). CDMA is in constant contact with all phones, coding each transmission differently (CODE division multiple access). TDMA phones give each phone a time slot and only talk to that phone (or listen to it) during that time slot (TIME division multiple access). Totally different technologies. Like watching TV on a toaster.
As for data on TMobile, you'll get 3G (for as long as TMobile keeps 3G going) - the phone lacks band 4, which TMobile uses for 4G. (AT&T seems to be using only band 17 all over now.)
Your best bet, if Verizon gives you the best coverage, is to swap your phone for a Verizon Note 3, or sell yours (
Swappa is a good place to buy and sell used phones) and buy a used Verizon model. They should run about the same price for the same model, so you'll be out shipping (a few dollars, probably). In a swap, you don't even have to swap cases, chargers, etc - everything on the outside of the phones is the same.