First, a quick note on technique - start with a coarse grade sandpaper on the floor sander and work down to the the finer grade.
The coarse grade removes the old finish and levels out the imperfections, then the fine grade makes it nice and smooth. If you go the other way around, you'll definitely curse every time you see the grooves in the sunlight. Fixing this means stripping off the nice new finish, not fun.
In passing, an obvious-in-hindsight but sometimes-missed point is to go with the grain, i.e. sand along the boards, not across them. In any case your sander will complain bitterly if you don't.
Also, give the boards an incredibly good wash before you start with e.g. Sugar Soap - you want to try and strip off everything gungy to stop the sandpaper clogging too often. Leave for at least 24 hours to dry nicely.
Be sure to hire an "edge sander" too, this will help you get to the edges that a drum sander won't get to. If you can, remove the skirting boards (baseboards) as then you can get right underneath; only do this if you know you're redecorating the walls too and don't mind patching up the woodwork. You'll inevitably end up doing the fiddly bits around the door frame and right in the corners yourself by hand with a scraper or sandpaper, which takes longer than you could possibly imagine.
While doing the sanding, remove everything from the room (obviously) and get heavyweight dust sheets to tape around the door. Wear airtight goggles as this is an insanely dusty process. You'll also need ear defenders as this is an insanely loud process too.
Regarding stain, it's a matter of personal preference. I've done both and prefer the more uniform result that stain gives.
Regarding finish, I've used both oil-based and water-based finishes. Both are OK, but the oil-based one looked richer (can't say why, can't say how) -- and took about three days to dry. Somewhat inconvenient for a hall, and our puppy left nice pawprints.
Follow the instructions on the tin; in general thinning probably isn't worth it, as you'll find the second and subsequent coats will go further anyway. I applied three coats of the oil-based finish, and five of the water-based finish, but this depends on the state of your boards. It's handy to rent good bright work lights so you can see where you've been, especially for the second and subsequent coats.
What is important is that you get flooring-grade varnish, regular varnish won't stand up to the first party, and yacht varnish will cost so much you won't have the first party...
Pine is a difficult wood to stain under the best of situations. Pine has areas of different density, color, and resin (pitch) content. Using a product like Minwax prestain helps a bit. Follow the timing instructions carefully when using a pre-treatment. Little tips: i like to apply a liberal coat of oil based stain with a sponge brush. Let it set a few minutes, then rub it in and evenly distribute with a lint free rag. As one rag becomes saturated with stain, I use a second drier rag to finish the rub in. The wetter rag is great for distributing the color. You don't want any really wet stain remaining on your wood after the rub in. Once you get to the urethane, sand lightly between each coat with 220 grit paper to remove any dust bumps. Remove all sanding dust with a slightly damp cloth, let it dry and apply next coat. Again, I prefer a good grade sponge brush, but a good grade regular brush for oil/urethane works good for appling the urethane. Remember, thin coats, and sand between each coat.
Best Answer
Do what a professional would do - don't touch it when wet.
Sand the affected areas with fine grit and re-stain, leave until it dries completely, then move, flip over and do the opposite side, etc. Apply the poly in the same manner. You cannot handle pieces until they dry fully. You also cannot handle them with hands that are wet with finish.