should i use a voltage regulator on a wall transformer.

Well I have to build a certain project, and so far i have been using a 9 volt battery. But I figure that a constant voltage source would be a better design. So I bought a 9 Vdc 150 mA wall transformer, yet when i checked the voltage with my voltmeter i measured 12.00 Vdc exactly. Well from doing some research online i found out that most wall transformers will give you a larger voltage at small loads. I’m not sure how much current my metronome uses, but i don't want to risk burning all those chips I used. So my questions are.

1. Should I simply measure the current of the circuit at minimum activity (least current usage.) and then use the wall transformer?

2. Should I use a 9 volts voltage regulator, to make sure i never over power my circuit.

3. If I do use a voltage regulator should i use a 1n4004 diode before the input? and should is use a .01 uFarad cap in parallel with the output voltage like I’ve seen in so many circuits on the web regarding voltage regulators?

I strongly recommend to use a regulator with 100nF capacitors. The regulator type depend from desired current, because when wall-adapter can't supply enough voltage on desired current, the regulator can't work properly. First measure the load current with another 9V supply. Test with same (resistor) load the adapter, and measure the output voltage under load. For normal regulator input voltage need min.2V higher as output.
If the voltage under test lower, apply an LDO regulator, it can work with 0.3...0.5V in-out difference.

What is the Capacitor for Again?

Thank you for your help. I do have a continued question. What is the Capacitor for. What can happen if i don't use it? And what type of capacitor? Electrolytic, Ceramic?

This is a ceramic decoupling capacitor, always nearby of regulator pins.
Recommended to avoid self oscillation.