For connectons, are you talking about something like a homerun? This is something where Verizon will install a splitter at your NID/enterance point that will filter out the DSL for the rest of your home. The DSL modem will be given it's own dedicated line.
If you don't have one of these, moving the router should be as simple as taking the modem/router upstairs and plugging it into a telephone line and power. If it syncs and your speeds are correct, it's probably good to go.
If you do need to change around any of the wiring, things depend on how your home is wired to begin with.