I assume this is all singer-songwriter stuff? Sounds like you want to look in that direction for inspiration. First thing I thought of when I heard this album is the Calexico / Iron + Wine album "In The Reins" (
sample track)
This is the sort of playing situation I am in mostly. What you're looking for is somewhere between modern and vintage. There are a couple ways to get what you're looking for:
1- Get an old vintage kit. I mean old - pre-war. The sound is not as sharp and the kick drums sound great. Oh, and they have a perfect vibe. I'd consider the sweet spot to be late 30s to early 40s; some 50s kits will get you there. Think Slingerland, Leedy, WFL.
2- Get a new kit with rounded bearing edges (eg DW Jazz or Classics are the ones I know of offhand).
That's the first part, and not necessarily mandatory, but it goes a LONG way to easing back on the harshness of the sound. Re-rings are totally fine *if* your edges are not sharp. (see Radio kings)
Next, heads. On your snare, you've got some options. Best 2 in my opinion are a coated ambassador or a renaissance. Leave it wide open. Tune medium to medium-low. Playing with rods or brushes is a great touch as well. A calfskin head is just a little too dull here. Wood snares are great; you can also try and scout up an old heavy brass drum. There are a few on ebay right now...
Toms - you could do calf, but be prepared to drop well in excess a hundred to head batter side only of two toms. I presume she's gigging so I wouldn't really suggest this route - only if you were about to record (and even then... heads aren't normally stocked, have unpredictable delivery and take about a day or two to dry up and seat). I'd do coated ambassadors top and bottom (or top and smooth white bottom) and just leave them on, let them age in and mellow. Don't change them unless you have to. Vintage As may work here, as would renaissance. Again: leave them on for the long run.
Kick: Coated or smooth white. No hole. Heavy wool felt strip. You want a heavy felt strip (a la ocheltree's old school felts), not those paper-thin things that Danmar and Gibraltar sell. Hit a fabric store and find the thickest wool felt they've got - cut two pieces about 2-2.5 inches wide, put one on each head. This is where an old kick would be helpful - you'll tend to get a much less pointed attack.
You'll be right where you need with the mellow, warm sound vs the sharper sound most modern kits get. As far as cymbals - think older. Either get some great cheap old thin cymbals from eBay or look at the Turkish manufacturers and stay thin. It'll get you that slightly trashy, more legato and less defined sound.