Our first rule of subpanels is Think Big. Really Big.
A 6-space panel might be dirt cheap and cure the itch today, but for a couple of pizzas, you can get a 30-space and cure the itch forever. We really want you to do that. There are plenty of stupid and useless ways to waste money in electrical work, but the one here is going too small and getting in the situatuon again after a few years.
It,s perfectly fine to feed a 200A subpanel from a 60A supply. You do not need a main breaker in this sub.
I want to see 30-40% of the spaces unused even with no use of double-stuff breakers, and between both panels about 48 spaces available for the typical house. That may seem absurd now, but believe me, if you have 48 spaces, it's a game changer. Stuff you've suffered with is now easily fixed.
I recommend using 6 AWG copper cable instead of 8. Thqt will let you run 60A instead of 40A. For such a short run, I would not fool around with aluminum. Wouldn't hurt, the lugs are aluminum, but copper is more flexible for the same ampacity. That may seem undersized for a 30-space sub, but the cable is easy to change when you need to upgrade.
If the existing panel is obsolete (Zinsco, FPE, Pushmatic) then I recommend a full 40-space for the sub, because then you're in a good position to phase out that old panel. That's more than you want to bite off today, but you'll thank me on the day you decide to tackle it.
Neutral and ground are rigidly separated at the sub. Any sub.
So you need separate neutral and ground bars, and you need to pull the neutral-ground bond (green screw or strap). You already have separate neutral and ground in your /3 cable, simply put it to good use. Provided neutral and ground bars should be a criterion in your panel shopping. Also consider "bonus breakers", some panel kits will toss in 3 or as many as 15 breakers.
Grounding rods unneeded
Since you are in the same building, you get to exploit the same, grandfathered ground rods. Your subpanel will get ground through the /3 cable.
Just remember, grandfathering only works if the ground rods are still up to the standard of when they were installed. If they're broken, you definitely want to fix them.
For any of the sizes you are comsidering (up to 60A), #10 is the ground size, but that is already in your /3 cable.
But my /3 cable is obsolete and has no ground
First, if you actually mean /2 and you plan to use the bare wire for neutral-ground, that is a no-go. Cannot do it.
However, if your cable's neutral wire is insulated, then you can do a cheat: run the cable inside metal conduit. The conduit inside diameter must be at least 138% of the widest point on the cable. Since the metal conduit provides a damage shield, there is no need to run up to the ceiling and across; you can simply make a straight beeline across the wall. The metal conduit, properly fastened through fittings, is a valid grounding path. Don't even try to bend conduit that large, just go straight across and stick spacers behind it where you strap it to the wall.