For the stairs, you'll want to rent a floor edger. First, take up the carpet. Then,start with a coarse grit (around 120) until you're through the varnish. Once you get down to the bare wood, use finer grits (up to 220) until the surface is perfectly smooth. For the banister, you'll need to use hand sanding pads. for all of the nooks and crannies.
When it's bare wood, use a mahogany stain and add additional coats until it's the color you want. Then seal with a floor-grade polyurethane.
Now for the door, you'll want to use a mixture of a random orbital sander on the flat surfaces (again, can be rented and follow the same course - coarse sand paper to fine) and hand pads for the detailed portions.
It'll take a while, but it'll be worth it. You can also check out chemical strippers and test them on inconspicuous areas to see if you could use one of those to get off the first layer of varnish.
Let us know how it goes!