Do you have a multimeter? For something like muscle wire, each different length you cut will have a different resistance, and you can't design circuits around it unless you know that resistance. Theory (eg if you know how many ohms per metre for your size wire) can help - but, if at all unsure, it is easier to simply measure it. Hence the multimeter.
Then you can just use Ohm's law, V=IR, to find out how much current it will draw when you put a given voltage across it - or, conversely, how big a resistor you need in series with it to limit the current to the desired amount. (hint - you will need a limiting resistor AND a drive circuit eg transistor)
Do you know the exact make of your muscle wire? Manufacturers usually write data sheets which tell you the stuff's properties, and they often include example circuits that can be very informative.