There are several different ways to mark the tailoring marks on a garment when cutting out. The one I always go back to is the one I learned first: tailor tacking. It's simple. It's quick. It uses items I always have on hand. And frankly, you tend to go back to the things you know.
Tailor tacking uses items that every single seamstress has. No matter how simply stocked your sewing stash, you always, always have a needle and thread. That's all you need. I prefer to use bobbins that were wound for other sewing projects. It gets those bobbins emptied off and ready for a new project and I didn't have to throw out the thread or take the time to unwind it just to empty the bobbin. Your needle choice depends on what you have on hand and your fabric choice.
A tailor tack is just a single stitch in the fabric. You leave several inches of thread on either side of the stitch. Leave it longer if you are going through several thicknesses of fabric (I like to leave four inches for two layers.) If you are marking an area you will handle a lot during sewing, it's best to leave a little extra as insurance against pulling out the thread.
After marking all the spots needed on the piece, remove the pattern and gently pull the fabric pieces apart. When you have about 2-3" of thread between the fabric pieces, clip the threads. Now it's ready to sew with. You can remove the thread as soon as you are done sewing that piece (just be sure you won't need that mark again before you pull) or just pull all the tacks out when completely done with the garment.
Tailor tacking works for most tailoring marks. All those little circles your supposed to match up, center line, fold lines, pleats, etc. are all great uses of tailor tacking. This technique does not work as well, however, with things like stitching lines, i.e. zipper flies and scalloped necklines. It's best to use transfer paper to mark those. You also need to keep your fabric in mind. If you are sewing with fabric that is easily snagged, it might be best to not be putting an extra needle near it. Heavyweight fabric like extra thick canvas or leather isn't great for this either. Leather just doesn't need the extra holes and you can hurt yourself trying to run a needle through thick fabrics. Transfer paper or pens are useful with these fabrics as well.
If you are new to tailor tacking, you might want to leave your pattern pieces around so you can double check which marks are which. The thread is just thread so it's up to you to keep track. It may seem confusing at first, but you catch on pretty quickly and the marks are seldom complicated anyway. I almost never have to refer to my pattern piece to interpret a mark. If I do, 99% of the time it's because I'm working on a UFO without the factory instructions and I've forgotten what the mark was for.
I hope you learned a new technique and you find it helpful in your future sewing.