You want to start with a chemical stripper. Methylene Chloride is the most common, fastest working, strongest, and unfortunately, most noxious. Put it on and let it work. Do a few steps at a time. If you have nice weather, open the windows for a breeze. The biggest problem people have is not giving it enough time and then having to spend more time and effort on a second, third, fourth ... application. Scrape off with a putty knife and follow up with a scotch-brite stripping pad. Neutralize / rinse as directed on the can of stripper. Then let the wood dry thoroughly. This will remove the polyurethane and generally most of the stain. Oak, having coarse pores in the spring-wood, might retain some of the stain in these rings. Don't kill yourself trying to get it out, just leave it as a contrast. Often people think they can just "sand off" the finish and avoid the chemicals. Bad idea. After a whole weekend and a pile of gummed up sandpaper, and only part way through the project, they think it's a bad idea, too. The other risks are that the finish is removed unevenly (causing the subsequent stain to absorb unevenly) and if there is veneer (not likely on your steps, but highly likely on casegoods), it's easy to sand through. An exception to this guideline is if you are doing a hardwood floor with a drum sander and removing significant amounts of wood with 36 grit belts. Once dry, do a light sanding with a fine sandpaper (120-220). I think you might want a stain, something like a "golden oak" if for no other reason than to even out the tones in the various pieces of wood. If you want to preview what the unstained wood with varnish only would look like, simply wipe some areas with a rag dampened with mineral spirits. I would also recommend applying a good floor-quality polyurethane. This is one instance where you need the abrasion resistance of a poly. Get a good brand, not just a can of polyurethane at the grocery store. Just because it says "Great for Floors" on the front of the cans mean that it's the best choice. If you want, you can thin it 50-50 with mineral spirits, creating your own "wiping varnish" and wipe it on. You will need 2-3 times as many coats as a brushed on finish, but it's fairly foolproof as a DIY or beginner. I also like to sand between coats to smooth it all out, remove imperfections and dust nibs. If you are wiping on, every other coat should be OK. If you are brushing, be sure to thin it so it flows out better.

"Are you suggesting I paint the fireplace mantel, staircase, etc a totally different color like blue or green? " Yes,as appossed to re-staining because (A) re-staining is so difficult to do a lastng job,and (B) paint offer's unlimited shades and tones to work with. Matching colors is far easier than coordinating them,stain make's it even more difficult. Frankly I'm not certain what I'd do after seeing the home in person. I suggested blue and/or green based on your discription(which I believe is accurate) of red floors and orange elsewhere. Call major paint distributors to see if they have a consultant in your area. If not,consider asking a paid consultant to have a look or taking actual samples of wall paint,floor and other woodwork to their office. Alternatly,use a camera with higher color resolution. After hearing you are already pleased with blue gray in another area I'm confident you can hit a double or triple useing it again. A color pro might turn it to a home run or grand slam. In the interest of full disclosure,this is not my field of expertise,it's just something I injoy messing around with. I may also be underestimating your eye for color and your desire to accomplish decorating without consulting decorators. I do it all the time simply because I like the satisfaction of "making it my own" rather than doing it "according to the book".

ok. I have another post going....I'm not sure how to link it here...(and I'm a software engineer....lol) I'll figure it out tonight. But here is a prototype. I wanted to see how I was going to make it in speck before I documented how I did it. so here it is...a $3300 chair built to speck for $400

tricom black from sherwin is a good black. do them in gloss they will look amazing. fine paints of Europe would be my first choice for this type of project