It really all depends on if you will have park admission for that day. If you do then take the bus or monorail from the Contemporary to EPCOT, walk through the park and exit via the International Gateway (between the UK and France pavilions). The Boardwalk resort is just a short walk, or you can take the boat. If you walk you will need to follow the path and then take the bridge to your left. ESPN will be the first club/restaurant you come to.
If, however, you do not have park admission it gets trickier. You would need to take a bus to a park, water park or Downtown Disney and then transfer to the bus for the Boardwalk. In this case you will get off at the bus stop and walk up to the lobby area. ESPN is down the stairs, past the green and then to your right.
Either of these scenarios will take up a bit of time. (At least 1/2 hour and possibly much more.) If you have a car it would be much easier to drive yourselves between the two resorts. As you drive out of the Contemporary you will take a left. At the first light you will make a right. Follow that road to the end (not far...you will pass the TTC and the Poly) and then take a left. You will now be on a main access road. Just follow the signs for the Boardwalk resort. Oh, and if you pay to valet park at the Contemporary you will not need to pay again at the Boardwalk...just show them your ticket.
Hope that helps.