How do I develop a new support system after moving across the country?

Get involved. It is important to get involved with the local community to meet people after moving to a new area. A person can seek friendship at his place of employment, volunteer around town, or join a local religious organization or church (if one is typically involved with those types of things). A person can even sign up for some evening classes in a subject or activity he enjoys, and meet people that way.
JOIN SOMETHING. It is tough move away and not knowing anyone. So if you work meet people through work and invite them over. If not then join activities like clubs or other things that can get you out into the community. Volunteer somewhere you like etc.