#1 First, is that face frame on your cabinets one big piece over several cabinets, or do you have an individual face frame for each cabinet?
I'm pretty sure the answer is one big piece. The cabinets are contiguous and if you took the doors off there is no paneling or decorative pieces on the framing. Just flat wood.
Odds are that there won't be any of that paneling or decorative pieces. What you'll be looking for is a seam between each cabinet that would tell you if each cabinet has an idividual face frame. If you can, from the bottom edge, look for that seam or perhaps take a knife to it to see if there's a seam that's just been painted over so many times that it's disappeared.
Either way, since you're wanting something you can stain instead of repaint, you should be able to easily build new face frames. Doors you can either buy prefabbed or if you're liking the newfound "Hey, I did that!" feeling, you can make your own. Not cheap, but not as hard and/or as expensive as you'd think.
The face frames are easily built using either pocket screws or biscuits. The biscuits would be a better application in this instance, IMO. You may even be able to walk in and use stock lumber to create your face frames. You have several choices of wood to choose from. If you can get away with stock pieces, you might wish to go with oak, otherwise you could go with some nicer pine or cedar even.....best to look at what's available around you, since I know wood is more expensive for you in your state than it is for me here.
If you can make individual face frames, that's even better, as you can work on one cabinet at a time, and you can set up a routine or use a jig to get things to come out in a uniform manner, and I think you and your wife will get more excited as the project unfolds if this is the case.
I'd price out a dedicated biscuit jointer for this, as well. Getting a slot-cutter for your router will work, but it's not as nice and easy to work with as the dedicated tool, and setup is way easier (to me) than the router setup. Bosch makes a nice one, as does Ridgid and DeWalt. I've got the DeWalt, but if I were purchasing another one, I'd pick up the Ridgid. The Bosch is the only other biscuit jointer I've ever seen people recommend, which is why I added it, but for no other reason than that.
I'd also say pick up a cheap pancake compressor/nailer set to attach the face frames to the cabinet itself. Mebbe $150-200. Look for Bostich or Senco. You could attach the face frames with the biscuit jointer as well, but that might be more of a headache for you to line all that up and over the entire kitchen it'll prolly end up frustrating you more than spending the money on a nailer.
Lastly, on the drawer front (HA!

) is it possible that you can simply undo the hardware and remove the drawer front, or is the joint something that you'd have to duplicate (like a finger joint or dovetail)? The best scenario is that you can simply replace the drawer fronts, which means you'd just run something through your router. I'm doubtful that it'll be that simple though. I'd say try a chemical stripper and then sand and stain on one drawer to see what the results will look like. If it sucks, go with prefab, if you're happy with it, great. Perhaps ask the wife if she can give you a complimentary color that you can just repaint 'em if they're not stainable too.
See if you can hunt up some examples of building a face frame today so you can see what that'd entail, and also give you a better idea of what to look at when you get home too. It's an easy enough project that you can be VERY confident of pulling off an excellent result. The doors are a bit more complicated, and the drawers being the most complicated (probably) of this project, but you should be able to tackle everything with reasonable success.