You need to do proper prep on everything, not just prime and paint.

See if you can get some 120,150,180,220 grit screen from the flooring guys. Abrasive screen is better in this situation since you will be smoothing the surface rather than removing material.

It'd take a day or two to do all the screening by hand. Cut the screen into manage pieces. To do the stair risers, railing shoe, and other flat surface you can wrap a piece of screen around a sanding sponge. To do the turning on the spindles, rip narrow strips of screen and floss the tight areas. To do profiled lengths and tight corner use the edge of the sponge or wrap screen around a dowel.

When I am doing very detailed work, I like to have an assortment of dowels, craft sticks, and chip artist brush handles that I save before chucking the brush tip. I wrap all the stick and dowels with assorted sand or screen. I find a craft stick with abrasive only on the faces and clean, crisp side come very useful when sanding around spindles and such.

Regarding the handrail: It is not necessarily to sand down to bare wood. However, it is important that you screen the entire surface uniformly. Basically you need to remove as much oxidized/degraded coating as possible and provide a good surface for the next step. Remember to wipe everything with alcohol. You will be applying 2,3 coats of shellac on the railing to act as a primer. Next you need to source a tin can of Wood Kote Jel'd Stain; do some trial and decide your choice of application method: terry cloth, pad, brush. I personally like to wipe it on with a terry cloth to get a good uniform color base, then use a stiff hog bristle brush for graining. Its up to you to get the right match, so do test pieces, and try different methods, number of coats, even 2~3 coats then seal with poly then an extra coat and final poly coat.

Regarding everything else, I recommend an oil based enamel. if you go this way I recommend the use of flow additives; I like xylene, it improves flow and slows drying time a bit. In addition to flow additives I recommend Japan Dryer, it works in the drying oils in the paint to accelerate curing.

Invest in a Wooster Flaxen brush.

Complete the handrail first the rest, clean up is easier that way.