I fully agree with skipping the coarse grade and starting with medium. And with the advice to practice with fine first. What is hard to recognize as you begin with medium is that the fine paper stage will take a LOT longer. You need the medium grade to sand off the worst, but stop and go to fine BEFORE you get all of the old color stains off. Basically, when you switch to fine, you are going to try to sand off all of the "high spots" down to the level of the bottom of the fine grooves made by the medium grit before it. Since the "high spots" are really about 90 percent of the surface area, that's a lot of sanding. So start that process early, because it will take off all of the last bits of old discoloration that are included in those "high spots".
I used a drum sander for the major portion, and a rotary edge sander for the edges. Rotary sanders do NOT sand with the grain and so leave odd marks, so you want to keep their use to edges only. A big rotary sander will be much faster than a small orbital sander, though. But I certainly can see the need for a small orbital in tight spaces like stair steps and corners. IMPORTANT: when using a drum sander NEVER stop in one place. It will sand down a groove really quickly. Start it with the drum raised off the floor, slowly lower it down while also moving the sander forward so that it does not land hard, and does not sand in one place. At the end of the run, keep it moving and raise it from the floor before moving the sander to a new spot. IF you have a really bad area that needs a lot of sanding, consider doing the first round at a 45 degree angle to the wood grain, then proceed to sanding strictly WITH the grain for all the rest of the work.
As Marlin1975 suggested, adding a little solvent (maybe 10 percent) to the varnish to thin it may help. This is especially a good idea for the first coat of varnish, which you actually WANT to have penetrate into the wood somewhat, rather like a "primer coat". In fact, the very open and absorbent bare wood surface will rapidly "steal" the liquid solvent part of the varnish so that it dries too fast, unless you give it extra solvent. Subsequent coats probably don't need extra thinning unless you feel it is drying too fast and failing to "level" (smooth out small wrinkles or wavelets in the surface) before setting up solid.
You have not talked about dust control which is a huge issue. I did these things:
1. I was lucky - I could do the whole job BEFORE we moved into the house, so impact on the family was reduced. Not many have that situation.
2. Block off all air flow points. Cover doors with plastic sheet taped to the frames, close windows to avoid having the wind carry dust around. Block off all hot and cold air heating ducts so dust is not carried out.
3. Get a good vacuum and LOTS of replacement bags. Vacuum up after every session of sanding. Do NOT wait until the bag is full, or even close. The fine dust (ESPECIALLY when using fine grade sanding paper) plugs the pores of the paper bag and it stops sucking. Worst case is if you burst the bag and fill the vacuum's motor with fine dust! So change the bag often.
4. After the last sanding and vacuum, wait 12 - 24 hours and vacuum again.
5. Read up on making and using a "tack rag" when using vanish. (Common old furniture finishing technique) This is a cloth with a little varnish (for stickiness) and some clean solvent worked into it so it is damp. You use it to wipe over the floor surface BEFORE applying the new varnish coat, and it picks up all the really fine little dust still on the floor. As you use it the solvent tends to get used up or evaporate, so you'll need to add a little more from time to time. Store in a closed contained when not is use to avoid too much solvent loss. You may need to throw one out when it's full of dust and prepare another to continue.
6. I put on three varnish coats, and the recommendation at that time was to rough up the previous coat with a light scrub of the surface with a very fine steel wool, just so the new coat had a good surface to bond to. So I did that, then vacuumed up the resulting dust, then used the tack rag cleaner just before applying the next coat.