Having had ATT&T, Verizon and T-Mobile domestically I shall try to rank them based on my experience.

Domestic network (does the phone usually work)
1. Verizon
2. AT&T
3. T-mobile

Domestic network roaming, size of network and availability of service

1. AT&T
2. T-mobile
3. Verizon

Domestic cost

1. T-mobile
2. Verizon
3. AT&T

International roaming service

1. T-mobile
2. AT&T
3. Verizon (really does not have any)

International roaming costs (with some exceptions such as Indonesia)

1. T-mobile
2. AT&T

Verizon does not seem to be a viable international option as one must rent or buy a separate phone - per their own web page. I switched from Verizon to T-mobile solely for the international roaming and the less expensive service with greater benefits.
Examples: Nation-wide long distance at cost of local call, Free domestic roaming, large number of minutes, free weekends including unlimited long distance domestically.

My plan is family. $75 with all taxes and fees for 2 cell phones. Nation-wide long distance, 800 minutes split between the 2 phones. One can use 0 and the other could conceivably use 800. International roaming and SIM unlock. Free weekends, 50 alpha-numeric messages per month. Web available for $2.99/month fee. Knowlegable customer service. The only down-fall is that the local service needs improvement - dropped calls and service area black-outs.