Beth,
We are planning master bath remodeling right now. Mold is a huge problem for us and one of the goals of our reno is getting rid of mold and mildew.
Here is our plan: I like tiled showers. I will use a large scale porcelain tile. To reduce the amount of grout lines, we will install linear drain - it allows to use large tiles on the shower floor as well.
Also, I highly recommend to invest into epoxy or urethane grout - it is not porous and mold will not grow on it. Regular grout is a heaven for mold.
Caulk lines will be minimal with these solutions, only along the joints.
Framed shower itself is a great environment for mold, so this is not an option for us.
And finally, install a good ventilation in your bath - powerful and quiet Panasonic fans are not that expensive.

You must caulk at change of plane (ie floor wall seems with tile), no grout there.

Forget tile and grout for showers. Install a solid surface shower with coved corners. Mold and mildew cannot live on solid surface.

Or get a better fan?

-Babka

Thanks for the input!
I will definitely invest in the Panasonic fan and urethane grout.
I went tile shopping today and in the course of the conversation, the salesperson said that some of my moldy caulk problem may have been caused by DH recaulking when the joints weren't completely dry. If moisture was a contributing factor, at least it's a non issue for the remodel. The frameless glass door is looking more promising.

If the joints are not dry the new caulk will not stick.
Use clear caulk.

I would highly recommend a solid surface unit as well. If you seal it up with 100% silicone you will be fine. Silicone is inorganic with no food value so mold and mildew cannot grown on it. The other big thing is make sure the silicone bead is applied 'smooth. Apply silicone, spray with rubbing alcohol, and then smooth out with finger. You can thank me later :)

Here is a link that might be useful: Solid surface showers

Thanks for the tips.
I know I'd like solid shower in terms of maintenance, but I was hoping to gain a couple of extra inches of width by tiling,

@schamltzy - Here is a quick video on how to work with silicone. Hope it helps you. Cheers!

Here is a link that might be useful: Working with silicone