How would I start to repair this Shurflo 2901-0213 water pump?
There isn't much to go on but I'll give it a shot. You say you switch the pump on and don't get water pressure. Given that you're trying to diagnose using a multimeter, I will assume the motor does not run. Between the leads to the unit and the motor there is a pressure switch (#1 on the parts diagram). It could have failed. 

More likely though is that the motor has failed. The motor will probably cost more than a replacement unit. (I was a yacht mechanic for 10 years and have tossed out more dead water pressure pumps than I can recall. No one makes a really good unit.) 

If the motor runs when you switch it on then there isn't much you can diagnose with a multimeter. Open up the unit and look at the valve assembly. There is probably dirt holding open one or more of the valves or one or more are dried out and don't seat well enough to function properly. A rebuild kit (new valves) will fix it if the problem is from bad valves. Otherwise save yourself a lot of grief and replace it.