That is an old "rule of six" panel, which while grandfathered, is illegal under its grandfathering becuse it has 7 main breakers. Going to five is a good plan.
It is a classic "CH" panel which is a very good industrial grade panel, except that the 3/4" breaker width make non-ordinary breakers very expensive (a trait it shares with Square D QO). That makes it perfect for what you plan.
On your subpanel which would be near this panel, I would get a panel with a main breaker, with an eye toward (at some point in the future) cutting it over to be the main panel. In a subpanel, the "main breaker" is nothing more than an on/off switch, it is OK for it to be larger than the feeding breaker.
I would also get a rather large panel, at the very least 42 space and even 60 or 84 if practicable: because panel spaces are dirt cheap and often even come with free breakers, whereas running out of space is painfully expensive.
I would aim for an industrial grade panel of good repute (one available in 3-phase variants, not Homeline, BR, or second tier brands) and avoid the expensive 3/4" breakers (not CH or QO).
Over time, as you find it convenient, i'd migrate all your 1-pole and smaller 2-pole circuits over to the new panel.
For your garage panel anything would do, but I'd go for the same type as your indoor panel, so you can use some of those bonus breakers. Again it's false economy to scrimp on spaces, I'd go 20-30 at least.
Also, since garage spaces need to be on GFCI, consider getting a subpanel which has a "main breaker" which is GFCI, that way all the breakers in that panel would be protected (at the cost of potential nuisance trips, a big deal if you keep a freezer in the garage).
Ed Beal raises some very good concerns about overall capacity. One problem with these "rule of six" panels is there is literally no main breaker to stop you from drawing more than 150A. So it pays to be conservative.
It's a difficult situation because you have two big loads that operate sporadically - the EV charger and the range. And the A/C as a wildcard.
One thing I might suggest, is feed the garage subpanel from the new primary subpanel. And then move everything but the range over to the new subpanel. At that point the only things still in the CH panel would be a 60A range breaker and a 100A subpanel breaker. Even at max, those two could not overload the 150A service (by enough to matter). This would force your entire house (from A/C to EV charger) to share 100A, but would remove the possibility of an overload. This would also save you the $85 you'll spend on a second 100A CH breaker.
What kind of closet will this be? Can't put a panel in a clothing closet.
A bathroom needs a 20A because a hair dryer is 13A all by itself nevermind the curling iron or hair straightener often set out to warm up while it's in use. Also whatever else, and lights, fan, heatlamp if that's on the circuit too.
I get your idea that it's YOUR house and YOU can cheap out on the electrical system and use your smarts not to overload it. However a) don't overestimate your sense of load size. b) I certainly hope for your resale value that you are not the last human who uses this house. And c) I fail to see the value of cheaping out on electrical. Given the astonishingly low cost of the parts, you should be optimizing for performance not economy. Case in point:
The ampacity of the subpanel
First, separate the supply breaker from the subpanel proper. You can have a 30A breaker in the main feed a 225A panel (the 30A certainly won't melt the 225A panel!) The reverse is not true, obviously. You would go with a higher amp subpanel mainly so you have extra breaker spaces for future expansion, because running out of spaces is a nightmare that is so easily averted right now, when you buy the subpanel, and extra spaces are cheap. So in your case I'd go for a 30 space for instance. Finishing with 60% of your spaces unused is a good place to be. It's so easy and cheap to just add circuits if you have the panel space, so things like under-sink water heaters, electric car, welder, become easy decisions.
Beware of double-stuff breakers. Many panels advertise "24-circuit (12-space)" - the 24 circuits relies on double-stuff breakers, which are more expensive. Those are bad news because increasingly you are required to use exotic breakers like GFCI, AFCI, 2-pole and the like, and those cannot be had in double-stuff. So you are stuck with the lesser capacity e.g. 12 circuits. It's cheaper to get the number of spaces you want, so you can use the simplest breakers allowed.
Which panel? The breakers must match the panel. Eaton CH panel, Eaton CH breakers. Most manufacturers make several incompatible lines (CH/BR, Homeline/QO, etc.) The ruling factor is the cost of exotic breakers such as GFCI, AFCI, 2-pole GFCI and the like. So you can cross off several models. Those exotics are being forced down our throat, so get used to them. I am also not a fan of tenant-grade cheapies (Homeline, BR). Siemens has a lot of promise: quality panel with sanely priced exotics.
The panel should be fed with a 2-pole breaker in the main panel, unless you are doing something intentionally weird, but then why would you use /3 feeder? You should know this, by the way.
I'm a little concerned with a knowledge gap here, and encourage you to hit a library for a book on wiring houses, and read it cover to cover. You don't want a knowledge gap. You especially don't want one after the drywall is up but before inspection. If your skill is lacking, do the works in EMT conduit so it's easy to fix later!
The circuits
Now, you say you want to run /3 cable. Either you don't know what you're saying, or you intend to double the circuit so you get 2 circuits out of each cable (a multi-wire branch circuit). That's awesome. (MWBCs not unfamiliarity).
I was a little worried that you were underpowering things like the bathroom and kitchen, but if you intend to supply them with a /3 and MWBC, that means double the circuits, and the cook will thank you. However MWBCs absolutely require a 2-pole breaker, which means you will need 2 breaker spaces for each one. Like I say, you go through spaces faster than you expect!
Honestly MWBCs are falling out of style because so many circuits require GFCI or AFCI. And that requires either the uber-exotic 2-pole GFCI/AFCI breakers, or correct branching at the destination and then GFCI or AFCI deadfaces or receptacles. And the savings is pretty small compared to running 2 separate /2 cable runs and avoiding MWBCs altogether.
The bathroom(s) require following one of two strategies:
Each bathroom gets a dedicated 20A circuit for ONLY loads in that bathroom, including wired loads like lights, fan, etc. I am not a fan of this one, because BOOM you are now in pitch black, holding a red hot hair straightener 1/4 inch from your ear, and can't even set it down because you can't see to not set it in the sink full of water. (however if the hardwired loads together exceed 50% of circuit capacity, looking at you fan-heater, that circuit cannot serve any receptacles at all, and this method doesn't work.)
A dedicated 20A circuit serves only receptacle loads in any number of bathrooms. The hardwired loads are served potentially from other circuits that serve other rooms.
There is one circuit that you really will want to be 100% dedicated, and that is the refrigerator. Code requires a GFCI since it's in the kitchen, even though that's not really the use-case GFCIs were intended for. And you really don't want any other appliance tripping the fridge and ruining your food. For this reason GFCIs on fridges are a terrible idea. I say install a dedicated circuit with a single (not the usual duplex) receptacle, so only the fridge can possibly be plugged in. Use a GFCI breaker, and I would certainly never tell you to switch it to a plain breaker 5 minutes after the inspector walks out the door.
Best Answer
Your main panel has several spaces open! Hurray! That means you have no problem adding a double-breaker for the subpanel. It also means that if you start adding more circuits inside the house then you are not in immediate panic mode. If you do start adding more circuits - e.g., a kitchen remodel - then you should consider a subpanel rather than going to double-stuff/tandem breakers. But no rush on that - even with the planned subpanel added you will still have 4 spaces open.
Conduit is great because you can change the wires later if needed. Rigid metal is even better because it takes care of the ground - one less wire to run.
The subpanel can be any size you want. But bigger is better, because you may decide later that you need a bunch of additional circuits for "reasons". The subpanel can be a "main panel" that comes bundled with a big main breaker (e.g., 100A or even larger), because that main breaker is used purely as a disconnect switch and does not have to match the power supplied to the panel. Your main panel is 30 spaces - a 24 or 30 space, or larger, subpanel would be fine.
The double-breaker in the main panel feeding the subpanel must match the wire. You could probably get by with even a 40A breaker & 8 AWG wire, but 60A & 6 AWG would be more typical and allow you to add a few more circuits without upgrading breaker & wire. The key is to make sure your conduit can handle larger wire in case you ever need it. 3/4" conduit would do fine for 6 AWG, but if you go up to 1" then you're good for up to 2 AWG and > 100A.
You will need ground rods at the subpanel.