How to calibrate monitor colors to use 3D glasses?

Most monitor calibration tools - the ones that come with devices that read the screen - are quite expensive and probably overkill for what you want.

There are guides out there - this one for example, and I know that you can get test cards to calibrate the brightness and contrast. You could try these first.

Also bear in mind that the free glasses that get given out won't be that accurate anyway.

Another thing to consider is that if you're slightly red/green colour blind or have one eye significantly more dominant than the other you might not see the 3D effect any way.

If you go to see a 3D movie can you see that OK? I know that uses polarised light and lenses but would rule out the dominant eye issue.

If you have the time (judging since you're trying to watch Youtube in 3D, very probably), you can do manual calibration.
Start by drawing a red (#FF000) square in your favorite image editor surrounded by white, and play with your monitor settings until the red square disappears. Do the same with blue (#0000FF).
That may be the easiest way, if not the fastest.