The best way to check a battery is to load it down, say 20A or higher, and then check its voltage. These days, a descent pack will be at or above 1.2 VDC per cell. A pack can have good "open circuit" voltage (no load), but it may not be so good under load. This would indicate that the cells internal resistance has gone up.
An esc can be checked with the same type of test. You would put one lead of a voltage meter on the Negative motor wire and the other lead on the the Negative battery wire. I am assuming this is a brushed controller. You would then put the car up off the ground and hold full throttle. There should be little or no voltage reading on the meter. Even loading the wheels should not produce much voltage on the meter. At 20A with an internal resitance of 0.0003 ohms you should only see .006 volts.