Gypsy, I hope you don't mind if I break this down for you in more simplistic terms. Everyone here is so used to talking about plans and such that they are starting the beginning (to them) when in reality they are starting in the middle (for you). This used to me a huge problem for me in the past, so I get it.
At home you have 3 separate systems.
1. You have Centurylink Home internet
2. You have Xfinity Home Cable TV
3. You have Xfinity Cell phone service
Numbers 1 and 2 are Sticks and Bricks ONLY. Disregard those. The fact that Xfinity TV was solid for you has no bearing on their cell phone services. Cell phone service is a completely separate business and is in no way connected to their TV system. The fact that your cell phone service has been solid has more to do with the carrier they have aligned themselves with.
On the road you will need Cell phone service. There are many Companies to choose from, but really there are only 3 main companies that have their own systems and networks. Those are Verizon, ATT, and T-Mobile. Every other company that you see out there with cell service uses one of those big 3 services to handle their cell phone duties. This isn't a bad thing.
Understand that each network has different areas on the map where you get better or worse coverage. Better or worse speeds, and so on. As HDR was explaining, if you choose from the big 3 companies, your traffic over those networks is their priority. Sometimes it's noticeable, other times it doesn't make that much of a difference.
To surf the internet, you need DATA, which will come from your CELL PLAN. As others have mentioned, each plan has different data caps/limits for regular use. You can also watch tv shows on your phone using data as long as the speed is fast enough. People here can help explain different plans in more detail once you know the basics.
HOTSPOT is a term for using your phone or a separate device to allow other devices such as your laptop to use your CELL PLAN DATA. Hotspot data cap/limit is usually not very large compared to what you are used to at home where you have no caps/limits. HOTSPOT DATA is something others can explain more about.
Guys and Gals, she's very new to the concepts that we now forget weren't second nature to use start from the beginning. We call the ABCs for a reason. Nobody ever calls them HIJs.