Hydrostatic testing is exactly what it says. You are testing with water that is not moving. Basically, the system is pumped up to 200 psi, or system pressure +50 psi, whichever is greater. You then monitor the pressure drop to determine any leaks. As Stookey stated, this is only done at initial install, repairs, or modifications.
Flow testing is flowing water through the hydrants. This will require at least 2 hydrants. You put a gauge on one of the hydrants to take your static pressure. You then open another hydrant for flow. This hydrant should run a few moments to make sure debris has cleared. You then place your pitot tube in the center of the stream and record the reading. Before you shut down the hydrant, you need to record the residual pressure on the gauge hydrant. Restore the hydrants to the original condition, closed with all caps on, and move to the next one.
Techinically, according to NFPA 291, you should open enough hydrants to get a pressure drop of 25%. I have yet to see anyone do testing like this.
Per NFPA 25, hydrants shall be inspected/tested yearly. Full flow tests of the underground piping is required on a 5yr schedule. If you are going to be doing any of this testing, you should have a copy of NFPA 25 and someone trained in Inspection / Testing / Maintenance of Water Based Fire Protection Systems (BTW, that is the title of NFPA 25).