My parents and sister all use Verizon Wireless since 2002. I personally have a Casio G'Zone Boulder v2/3 and have little to no problems, my parents usually handle the finances as my sister and I are still covered in the family package.
The only time I had problems was the January Blizzard that damaged over the local cell tower and when I was out in the middle of the ocean on a Cruse ship (dur). Reception is by far the best that I know of with very few dead zones (mountains mainly). I have little information on the internet or browser stuff, as I just mainly use my phone for calling home, texting, and emergency contact phone for the National Guard. I also never seen my phone go into roaming unless you were out in the middle of the desert.
They have a broad range of phones as you would expect with a major phone carrier and usually carry the better brands (with the exception of Apple I believe, but I won't touch an iPhone with a 10ft stick).
You are allowed to upgrade your cellphone at discount at the end of your two year contract. Verizon Wireless is known for there higher prices and military discount is kind of funky (very different than other cell providers).
Customer Service is pretty good. When my parents struggled with some payments, they understood and gave some leeway until they caught back up on the payments (you just have to communicate with them).
Would I recommend them? Yes, I have used my phone all across the US with no problems (Washington, DC, Chicago, Oklahoma, Texas, Arizona, New Mexico, California, Oregon, Nevada, Utah, on and on.). I have not been able to test Verizon overseas. The only downside is they are pricer than say Sprint Nextel or Altell.
Pros:
Good Coverage
Broad Range of Products
Cons:
Pricier
One recommendation, depending on how active you are and not worried about rapid texting, is to buy a rugged phone such as the G'Zone Rock. They are more expensive, but they are guaranteed to last.