There are two major providers in Arizona. Cox and CenturyLink.
Cox is a Coaxial provider, and provides Internet connectivity up to 1 gigabit over the lines you get cable on.
CenturyLink is a DSL and Fiber provider.
Verizon, T-Mobile, ATT and similar can provide home internet via cellular signal, similar to the internet you get on your cell phone.
Cox is always my recommendation unless you can get CenturyLink Fiber. They provide the best customer service and are more stable and faster than CL DSL. If Cox is not in your area, CL is still better than the cellular providers. Cellular providers, due to not being wired and having the lowest caps on usage are the last resort if you cannot get Cox or CenturyLink.
The internet is fragile and no provider is perfect. No service will work flawlessly all the time and many internet issues we see are not related to the actual provider, but rather the in home tech, despite the constant blame on the provider. Just remember not to use the ISP provided router.