Hi Steve, and Welcome to the forum!

It looks like you have done a pretty good job so far, with good sized rooms, and making good use of that strange-shaped space.
The construction is inside-out, room within a room, made of 2x4s, 2 1/2 sheets of sheetrock with Green Glue, and Rockwool insulation.

You should get pretty good isolation like that.
I realize that the window in the back of the control rom is an odd choice, but it will be covered with a diffuser most of the time, and is only there because I have some situations where I will need to see in that room.
With 17' length, your room is probably long enough to be able to use a diffuser back there. Many small control rooms aren't really big enough to be able to use diffusers, since you need about ten feet between the face of the diffuser and your ears to be out of the "danger" zone, where the diffuse field is not properly smooth and even, but you seem to have enough space to do that. I imagine you are going to do some type of "diffuser on wheels", that you can move over to one side of the room when you are nor doing critical listening, then wheel back when you are mixing/mastering? That can certainly work. Of course, the room won't be accurate when the diffuser is wheeled off to one side, but that's likely fine for typical tracking sessions. One suggestion here: I'd really recommend going with a Tim Perry style "Leanfuser" for the rear of your room. I've had good success with those: I've taken his concept and adapted it to a few different rooms, and it works well. In your case, you could probably build it in two "halves", split down the middle, with each half on its own set of wheels, then still keep some semblance of symmetry by having each "half" wheeled off to a similar position on the sides, symmetrically. You might even be able to use those "halves" creatively in the tracking room...
Also, the double set of glass doors at the entrance are mandated by the building, and cannot be changed, hence the small "lobby" in the model.
That's fine! And you might even be able to use that space as a third iso-booth for some overflow situations. The acoustics would be pretty lousy in there, but the space itself is still usable. For example, in a pinch you could put the bass player in there, or the keyboard player (if you can fit a keyboard in there): those are not acosutic instruments, so you don't need to mic them, and the musicians can listen on headphones.... Not ideal at all, but still usable! So you might want to put in a few connection jacks in that area, just in case.
In general, I like a fairly tight control room, and a fairly lively live room.
For the control room, the optimal acoustic response is specified in several documents, such as ITU BS.1116-3 and EBU Tech.3276. You can find both of those here:
Useful reference documents for designing and building a studio The basic concept for control rooms is: neutral. Transparent. Not there. In other words, the control room itself cannot have any effect on the direct sound from the speakers. It must allow that direct sound to reach your ears, and ONLY the direct sound. Nothing else. The room must not add anything to the direct sound, nor take anything away from it: just transmit the sound cleanly, absolutely unaltered, from the speakers to your eras. It's very obvious, when you think about it! It's one of those "Duh! Of Course!" moments for many first-time studio builders. That is, indeed, the entire purpose for even having a control room at all: so you can hear the speakers, only the speakers, and nothing but the speakers, and you can hear them "pristinely". It's easy to see, and easy to say, but not so easy to achieve in practice. And that's what those documents are all about: setting the acoustic specifications for the speakers and the room, telling you how it should be for the best possible critical listening... but they don't really tell you HOW to achieve that! They just tell you what the goal is, without revealing the details of how to get there.
The amp room is small enough that I believe that I will need to keep it pretty dead just to control the sound, but I am willing to take advice on that.
If you only ever plan to use that as an isolation booth for your amps and cabs, then that might be an option, but even then you'd probably want to leave it a bit brighter than "totally dead". That gives you more options with mic'ing those amps, as you can then go for some "room sound", or "ambiance", in addition to just having a close-mic stuck up against the grill. With a pleasant sounding room, you can have a second mic further out, then blend those together in the mix, to get a "fuller" or "warmer" sound, that is more natural than trying to do the same with a plugin or reverb box. You could even go the whole nine yards, and make the acoustic response variable, like this:
What is variable acoustics? How do I do that? giving you an even broader range of acoustic response, with greater flexibility for different situations. For example, if you ever wanted to track brass, other large wind instruments, or large strings, such as cello or even violin, or maybe sax, then you could use that room as an instrument iso booth, adjusting the variable panels, the instrument location, and the mic location, to get whatever sound it is that you are looking for. A well designed set of variable panels can give you very dead and dry with one arrangement very bright with another setup, or perhaps diffuse... and a large range in between, for intermediate positions. The possibilities are endless. And once again, a natural room sound usually beats a "canned" sound from a plugin or black box. It's also more pleasant for the musician, to play in a room that has some life and character in it. I do realize that this room is basically just meant to be an amp iso booth, and that you have a better, larger live room too, but I'm thinking more for the occasions where you need to track several things at once, and want some acoustic separation. I would even suggest putting a small patch panel in your green room, so you can use that one too, as yet another "booth", for musicians with non-acoustic instruments. That would give you a potential 5-room studio, if you count the mini lobby and the green room. That's pretty neat, when you have to track a large band all at once!
Just a suggestion, of course!
I left the amp room just a rectangle, as I am guessing that I will need to do some angling, trapping, etc. based on the acoustics.
The supposed need for angles is a myth! I cover that here:
Angled walls: What are they for? Do you need them? That's mostly about control rooms, but also applies to live rooms. Basically: angling walls, or even surfaces within a room, does nothing at all to deal with the acoustic problems in the room. What it DOES do, is make the room smaller, and therefore WORSE, acoustically. There is nothing at all wrong with a rectangular room, that is then treated as needed to get the response under control.
My question at the moment is: how do I proceed with testing the acoustics of the rooms. I imagine that I need some idea of what I am working with before I start with resonators, traps, etc. Any advice would be greatly appreciated. Thanks!
Starlight already pointed you to the REW tutorial, so that should help you with the testing. That is mostly about using REW to test/tune control rooms, but some of it is applicable to live rooms.
But before you can do that in your control room, you first need to get the layout settled. Sometimes called "geometry", that refers to the geometric relationship between the speakers, the mix position, and the room itself. I talk a bit about the famous "equilateral triangle" method for doing that, here:
Speaker setup, and the equilateral triangle That might help you to work on your initial layout. There's also the "walking mic" test procedure, that can help you find the optimal location for your mix position, and it's not hard to do the same thing the other way around, "walking" your speakers a bit to find the best location for them. This is a slow, tedious, boring, process! Lots of minor twiddling and tweaking mic and speaker locations, mountains of REW tests, but it is guaranteed to point you to the best possible geometry for your room... then, based on the final REW test once you have that geometry located, you can start on the design for the room treatment. Everything you need to know will be in that final REW test (OK, maybe not "everything", but at least all of the major stuff).
The only additional comment I'd add here, is a very strong recommendation that you consider flush-mounting (a.k.a. "soffit mounting") your speakers. That is, without any doubt, the single biggest and best thing you can do for any control room. Flush mounting your speakers totally eliminates or greatly reduces the gross artifacts associated with having speakers in a room. It removes them from the room, thus also removing the main artifacts. That would be the number one recommendation.
#2 recommendation: minimize your desk. Right now, you are showing a large desk very close to the front wall. Large desks have a significant negative effect on room acoustics. Make it as small, as low profile, and as "acoustically transparent" as you possibly can. This thread might help you a bit with that:
The Soundman M1 studio desk.
Overall, it looks like you have a nice place there, with strong potential. I think it can be
very good, with careful layout and treatment.
- Stuart -