Teak (as most woods) has soft and hard parts of the grain. If you scrub teak with the grain, and use the chemicals (as ghone suggests)--brightener will have oxalic acid, and if you use a stiff brush can actually remove part of the soft wood--leaving voids in the surface of the wood. Same with teak decks--one should brush across the grain, and when possible avoid the harsh chemicals.
You can use very mild chemicals such as "spick and span", next up the harshness scale is "Bar Keeper's Friend" which has both oxalic acid and a mild abrasive. Finally are the multistage cleaners, with higher concentrations of oxalic acid and sodium hydroxide. Serious stuff--and I have come to avoid these.
If you elect to sand, then either by hand, and do not use harsh grits (start with 320 and shed how that does---work up to 600. If using a sander, consider a high speed orbital--again be very careful about getting any cross grain "cuts"--they have to be sanded out.
As to finishing teak: I do like to protect them. I have tried all types of oil, the latest and greatest varnish, even epoxy, pigmented (Simco) or Cetol. All in all , I have concluded that going back to a good spar varnish is probably the best solution. Leaving it raw, or oils tend to detiorate more rapidly, and in the long run is more work that a good varnish job. For varnish, be sure that the teak is dry, vacuumed, and then tac ragged. The first coat, I like to thin 50%. A very light sanding with 600 between coats. Use a very high quality badger brush---or go with the cheap foam applicators. Don't use cheap brushes. If you are going to the trouble to do a good job, I think it is worth the while to take the rails off the boat, and put on at least 6 coats of varnish. I have see epoxy last well, if coated over with urethane varnish, which has UV inhibitors (We did this for the teak and holy sole in our Cal 46, and 10 years after, it was still like glass--just a new coat of urethane every 3 to 4 years.)
After the rails are refinished--and back on the boat, you may want to make some simple Sunbrella railing covers to protect them.
One advantage of removing the railings is there is no masking issue on the deck, and you can make the sanding and cleaning smooth and even near where the railings meet the deck.