How do I update a tile bath/shower combo inexpensively?

if you are keeping the tile then you can have it refinished white or any color you desire. You could also change the tile out with larger tile pieces and maybe add some mosaic tile to change it up. If none of these work then have someone take it all out and replace with a bath tub insert. It is limitless with all of the new products out there. Good luck.

Why not try reclaimed wood on the sides. There are tile refinishers and DIY paint made for tile but they don't last forever inside of a tub. Leave the inside of the tub grey, and tile over the black with a new color tile. You CAN tile over tile. Cover the top lip and front of the tub with wood, bead board would look pretty with a trim up top. Tile over the black front tile or liquid nail a pretty wood trim over the bead board facing. It would brighten and add a new color.

Rustoleum has tile products that can change the look of tile. I came across this idea recently of putting beadboard across the front.

https://www.hgtv.com/design/rooms/bathrooms/update-a-bathtub-surround-using-beadboard

You don't, if you want it right take it off and replace it, don't always look for a cheap way out

call a sink and tub refinisher and have them refinish/paint the whole thing in color of choice. the end product will have the same finish as a porcelain tub and be uniform in appearance.

you could also retile in wood looking tile for the front[keeping it all waterproof] and paint the rest , but the refinisher would look more professional

I would call BATH FITTER, THEY ARE THE PROS.

If you realy have to save, then remove broken or damaged tiles, then remove in a random way half the tiles & replace just them with decorative ones in colors you love! Grey goes with anything, burgundy, green or black. I would make some of them with a pattern or design on them. Good luck!

You could do some stenciling around the tub/wall, there's so many different stencils to choose from. Just make sure to seal it afterwards.