What grit sandpaper should I use between coats of polyurethane?
Read the instructions on the can.  See what the manufacturer recommends.

Sandpaper is fairly useless on polyurethane because the heat generated by the friction causes the polyurethane to melt and block up the grit.  And the finer the grit, the faster it gets blocked up.

I have been using polyurethane for more than 30 years.  Here is my routine to achieve a fine finish:

Note: If you do not have a dust-free environment, you should try to apply the polyurethane outside on a calm day.  Dust particles falling on your work are a major problem.  Stay out of the sun, though.  


Sand work piece surface to 220.
Vacuum work piece surface.
Using a soft cloth, rub a very light coat of mineral spirits into the work.  This will give the polyurethane a head start penetrating and “gripping” the wood and will retard the drying time slightly.  This will also allow the grain to appear more vividly so you will get some idea of the finished product.
Wipe down the work piece with a tack cloth and immediately begin applying polyurethane.
Apply polyurethane with a fine brush, working across the grain first, then with the grain.  Work quickly, overlapping and moving the polyurethane along the piece.  Apply only a thin coat.  Do not attempt to re-brush the polyurethane once it has “leveled.”  (When the brush strokes disappear.)
After drying, sand first coat lightly with 220 paper on a sanding block.
Wipe off polyurethane dust with a soft cloth, then wipe with a tack cloth, and begin applying second coat.
I do not use sandpaper on the second coat.  I use medium steel wool on the second coat, and fine steel wool on the third coat.
After a few weeks, you can apply a light coat of carnauba wax using very fine steel wool or just a soft cloth.


Note:  Never work directly from the polyurethane can.  Pour the polyurethane you need into a clean plastic cup, then work from the cup.  Discard any unused polyurethane and the cup.  Never pour unused polyurethane back into the can.  If you do, you will just be adding more dust particles to your next finishing job.