I know this may sound stupid, but WHERE do I put Plumber's Putty for new faucet?

I am Installing a new faucet that comes in 4 parts, each installed in seperate hole in sink: Handle, Spout, Sprayer, and Soap Dispenser. One of the tools that is recommended in the instructions is Plumber's Putty, but they do not specify where to put it. I am guessing that it goes on the base bottom of each part before I fasten them into place, but I am not sure, and don't want to do it wrong. Please help.

Yes, it would go wwhere youthought but depending on the brand there may be a rubber or foam gasget that goes down in place of the putty.

What material is the counter made of or is this being mounted on a sink?

I am not a professional but as already posted, it would depend on the brand and whether there is already some other form of gasket but I installed the same setup as yours and there were gaskets on the handle and the faucet so I needed to use plumber's putty only on the hose sprayer and the soap dispenser. I filled the whole "gap" in each device with the putty so that some was pushed out when tightened. Warm the putty by rubbing/rolling in your hands first and it will be easier to work with.

putty

It goes between the sink and the faucet part. Putty will damage some faucet finishes, but if the instructions say to use it, then it is okay.

Putty can also stain some natural and artificial stone counters...
Check the manufacturer of yours to see if this is the case and follow their recommendations...