Summary

Welcome to the pinnacle in Japanese sport-luxury beverage placement technology - the Cup Coaster 350 F-Sport.

The print is in three parts:
1) The base (the body of the coaster),
2) The insert (fits into the top of the base), and
3) The pad (fits into the bottom of the thick version base, not used on the thin version).

I've provided two versions of the base - a thin and a thick version. The thick version is a little taller and it has a cutout underneath where the pad fits. The pad is meant only to grip the coaster to the table. The thin version has no pad. I don't feel it's absolutely necessary, but my end tables are textured. I have no idea what your needs might be. But nobody wants a slippery coaster, so I thickened it a bit and made a bottom pad.
In other words - if you want the thin version, print the thin base and the insert, but if you want the thick version, print the thick base, the insert, and the pad.

Both the thick and thin versions are 4" wide x 4" long (or 101.6mm x 101.6mm), but the thicker is 10mm high, while the thinner is 7.8mm high.

The base is meant to be rigid, while the insert and the pad are meant to be more "rubbery" to act as a grip. For the base material: I personally don't print in PLA because I prefer PETG, but PLA+ should do if that's what you have. For the insert and pad material: I would only recommend TPU. I recommend gluing the pad onto the base, but the insert should fit snugly into the base and not need glue. I printed mine in a 0.20 layer height and that worked perfectly fine. There are no intricate details on the model.

I have found that TPU prints well, but the top layer is rarely perfectly smooth. This is a good thing for grip, but if you're a stickler for appearance then try printing the insert upside down. I use a textured bed and it both looks good and provides a more grippy surface.

If you want to skip the insert entirely (perhaps you don't own any TPU), that might be an option as well. I definitely recommend the insert, but I rested my coffee cup on the base while my insert was printing and I had no issues with it. I have a decent-sized iced tea cup that I put on it after I was sufficiently caffeinated, and that also worked well. But I am not sure the insert-less coaster would work well on smaller or more delicate cups.
Now that I think about it, if you print the thin base out of TPU, you wouldn't need the insert or the pad. Just a thought.

I created the models in Blender. I've included the .blend file for you as well.