First problem: those 10/2 cables ate your conduit fill
Your first problem was that for some reason, you used cables in this conduit run originally, instead of individual wires; since a cable must be treated having a round cross-section for fill purposes due to cable twist, stuffing cables down your conduit is what left you with no room for anything else. To wit: a 3/4" Schedule 80 PVC conduit has a usable fill area of 105mm2 (if it's 3/4" Schedule 40 PVC, or 3/4" EMT, it's bigger inside than the Schedule 80, but not by much), and a 10/2 W/G UF cable, measuring at 11.684mm on its major axis, takes up 107mm2 of fill area, overstuffing the conduit then and there!
Worse yet, if they're NM cables instead of UF (the latter is notable because its jacket is almost always a distinctive light grey), than they have to go anyway, as NM cable is not usable in wet locations, such as the inside of underground conduit (if you don't believe me, see NEC 300.5(B)). Using it there not only violates NEC 334.12(B) point 4, but leads to failures down the line as the paper separator in NM will wick moisture into junction boxes and drip, drip, drip it all over the electrical bits inside.
Once we address that though, this becomes a lot nicer
Once we rip those errant 10/2 cables out, though, this becomes a much nicer job. We can then fit 3 6AWG stranded copper THHN wires with a bare stranded copper 10AWG ground down this conduit, for a total of 103.4mm2 fill; while a tight pull, it's certainly not an impossible one, and much easier than wrangling those terribly miserable cables!
That gives us a nice, healthy 60A feeder for our money here, that we can then protect with a 60A, 2-pole breaker in the main panel. Once that's all straightened out, we can then move onto replacing the subpanel; since this subpanel is in a separate structure, we need a main breaker in the subpanel to serve as a cheap shutoff switch. Furthermore, 60A bussing is far too limiting in subpanel sizing, so we go to the next major size up, which is 100A or 125A. However, amperage is not as large a concern as the number of spaces the panel supports (not "circuits", which is rather a lie): 24 spaces is the bare minimum I would use in this application, with 30 space panels being preferred for this.
Grounding this all
First off, we need to make sure the subpanel has matching ground bars fitted to it, so there's a place to land the ground wires; with that done, we then can pull the bonding screw out of the panel (most likely a green screw, somewhere in the neutral assembly, the labeling or instructions for the panel will say where it is), which separates ground and neutral at the subpanel, as there can only be one neutral-ground bonding point in your system.
Last but not least, in order to return lightning-induced currents back to terra firma, you need to put in a grounding electrode system at the garage, as you allude to with your mention of a ground rod. For a system such as yours, a pair of 8' ground rods, driven 8' apart, and connected to each other and to the panel with 6AWG bare copper wire, will normally do the trick.