Grout in a change of plane (eg between floor and tub) is never a good idea - it is a matter of when it will crack, not if. The cracks between the tile in your photo, however, indicate an unstable subsurface, which may very well be from water constantly getting between the tile and tub over the years, and rotting the subfloor.

If you're pulling it all up, first repair/level the subfloor, then if possible, lay down an uncoupling membrane (schluter, wedi, or redgard each have products i believe.) This will make sure that movement of your subfloor doesn't transfer to the tile field. Some are suitable for installation of under-floor heating elements as well.

Next, while a bit more expensive and slightly harder to work with, i suggest a polymer modified grout as they are less prone to shrinking, bond better, and are more stain resistant (epoxy grouts are another option, but may be even harder to work with.)

And finally, don't grout between the tub and tile - that's what caulk is for. And not the cheap acrylic latex stuff - you at least need silicone, which you can often find from the same company that made your grout, color matched. An even more durable option would be an MS-polymer sealant/caulk, but you'll be pretty limited in color choices with that option.