To measure amps (current), the meter has to be wired in series with the item to be measured (for this reason, ammeters are designed to have very low resistance).
This has the down-side that you have to disconnect the component to put the meter in line with it. That makes it hard to do the "check a component at a time" method you mentioned.
An ammeter measures actual current flow, so you really can't test a component for it in isolation. Components can have wildly different "current draw" depending on the situation. For example, motor current varies with torque and speed; current through a resistor varies with the voltage across it; and so on.
There are special "clamp-on" current meters that just clamp around a conductor and report the current by using induction. Very nice if you have one.
If you just want the total current the entire RAMPS board is pulling, put the ammeter between the power supply and the RAMPS power input connection(s). Be very sure not to have the meter set to read volts or ohms when you do this (it might or might not survive).