Hi,
I've done both before (painted & replaced).
Painting - definately an easier/quicker option, with less skill required. We used the White Knight product (primer/top coat) and worked well. Good coverage, nice finish - gloss or matte. I wouldn't recommend it for high wear areas - like flooring - but if it's just getting a splash from the shower then should be fine. Probably worth double checking with the manufacturer. At the end of the day it will look good - but painted.
Replacing tiles - you can chip out the tiles - you need to be really careful, not only of the surrounding tiles, but also of damaging the sheeting they're stuck to. If you damage that then it needs to be replaced before re-tiling. Once this is done, you'll need to glue the new tiles in and try and grout with a matching color. You may also find difficulty finding matching tile color.
Unfortunatley, you can't buy a small pot of the tile paint (that I found), but if you're happy to spend the $40/$50 on 4L then you can always buy and try on some sample tiles first. This will be your cheapest & easiest option if you're happy with the end result.