I have never resealed a bath before. I'm confident on what I need to do except for one thing... A lot of the sealant is still in great condition. It's only the sealant behind the taps that has gone black. Do I still need to remove ALL of it in order to get a smooth continuous finish around the whole bath? Or can I just replace the bit behind the taps and it still look like the sealant runs smoothly around the entire bath?
Remove, clean, recaulk. Leave a small uncaulked spot on the bottom of the taps. Also, use a wet sponge to wet your finger, then drag along your fresh caulk, rewet, drag another section, etc., this will work caulk into crevice, smooth out the finished surface, and give a professional look. Just rinse your sponge out before caulk dries in it.
It could be quite difficult to match the colour, even if you bought the same brand the original sealant may have changed a shade over time. It would also be tricky to make the bead look continuous, some sealants are hard to work with. You could try just doing the complete end of the bath that the taps are on, up to the corner. That way you wouldn't notice any differences in shade so easily, since the bead turns 90 degrees. Of course I am assuming that the taps are at the end of the bath, if they are in the middle of the side then that is a lot more to re-seal.
The taps themselves may obscure any obvious joins in the sealant, so you might get away with just doing the bit behind the taps - in fact since it's a small area you could just start off doing that anyway and if it doesn't look good after it has set you can try an entire side. Wrap some insulating tape around the end of the sealant tube nozzle if it doesn't come with a screw on lid, to keep it fresh for a while.