First, the space you plan is just fine for this use. Wise to mount a 3/4" sheet of plywood (even better: something nonflammable) to the joists, simply to give them freedom to mount the panels, conduits and attachments as you please, instead of being forced to line it up to joists.
Most panels are 14.5" wide because joists are on 16" centers and 2x4's are 1.5" wide. This does not include the cover, which will overhang too much to put two panels in adjacent joist bays. I don't like in-joist mounting anyway, as it wastes the dozens of side knockouts and forces you to cram everything in the top or bottom. Ugly.
The subpanels will come with neutral buses. You will need separate ground buses - think about that when buying panels, some cheapie panels sell you the ground buses separately - costs more in the end, worst of both worlds! If that main panel is a "CH", that's a fine type, you can reuse your breakers.
As I mentioned, don't be bashful about making the first subpanel absolutely huge in number of spaces. 42-space is not excessive. The problem is right now, buying 10 more spaces will cost you maybe $30. Later, 10 more spaces will cost you a world of labor and finding a location for another panel. Wildly oversizing the first panel means not ever needing a second. As a bonus, larger panels come with more "bonus breakers".
Do not worry about the subpanel "amp rating" It needs to be greater or equal to the breaker it's fed from, but it's totally fine to feed a 200A subpanel from a 100A breaker!
Another trick you can do with "subpanel #1" is groom it to be your future main panel. In that case you'd buy a subpanel with a main breaker. It's OK for the subpanel's "main breaker" to be larger than the breaker it's fed from.
Your subpanels do not need their own main breaker -- they are in the same building as the panel they're fed from.
I would run nice, big, fat, 1-1/2" or 2" EMT metal conduit between the main and subpanel #1, probably from the bottom. For two reasons - ease of pulling and room for expansion. Use "conduit bodies" or prefab bends to round corners. The metal is the ground which means you only need to pull 3 wires :)
I would install #1 AWG aluminum which is good for 100A. Given the very short distance, I wouldn't object to the cost of #3 copper, but the lugs you'll be attaching to will be aluminum. Why create a dissimilar-metal problem when you don't have to?
I gather this power is for a server cluster or something that will draw most of its load continuously.
You want four 20A@120V circuits. Most people wire their subpanels as 2-pole 240V/120V. That would allow you to wire the four circuits evenly on two poles, so it would be 40A @ 240V.
Wiring the subpanel as 120V only @ 80 amps is not a great plan. It will create 80A of imbalanced load, which will create high stresses in everything in your electrical system including the power company's feed. Note the thing ThreePhaseEel says about downsizing neutrals - it breaks that. You will need twice as many spaces in your panel, as every other row will be dead. And your breakers will be more expensive.
Since you see this as a very limited use panel, I'm ok with at least a 12-space panel so you have some breathing room. As discussed, if you wire it as a 240V panel balanced, a 40A feed will suffice.
This can be wired with #8/3 AWG copper wire, but given the short distance, run #6/3 and you'll be able to bump it to 60A later. I would just use cable rather than conduit in this case, since you have an intermediate device to attach to. If running conduit through that makes sense, then sure, do EMT and let it be the ground :) But 3/4" conduit will limit you to #6 wire.
Subpanel #2 off main or subpanel #1?
I'm perfectly OK with feeding subpanel #2 off of subpanel #1. However, this will add 40 (-80?) amps of demand to subpanel #2 and let's look at breaker prices for a moment.
- 40A breaker - $9
- 60A breaker - $9
- 100A breaker - $45
- 125A breaker - $85 and 4 spaces
- 150A breaker - $135 and 4 spaces
Yikes. Feeding off subpanel #1 will raise the cost of its breaker quite a lot, because it has to passthru an extra 40A guaranteed, so its breaker will be $80 more. (now were you just mentally arguing with me about a 42-space panel being a needless expense? ;-)
I prefer to stay in "cheap breaker land" and feed sub #2 with a $9 breaker and sub #1 with a $9-45 breaker. (do you really need 100A if they are separately fed)? I would still run the #1Al/#3Cu wire so you are 100A-ready.