That is a GE panel. Use only GE thin breakers in it, they plug on the small pieces welded to the sides of the stabs, visible in the pic.
Is the neutral bar in the pic insulated from the box itself? (except for the flattened strap with the green screw in it). If so, go ahead, and buy a ground bar, a GE bar. The big box stores carry them hanging in bags in the electrical section. There should be holes pre punched in the back wall of the panel, the holes will line up with the screws in the bar. Install it, and run your ground wires to it when you wire up your outlets. Also run the single, continuous, uninterrupted wire from the ground rods to it. No splices or connections allowed in that ground wire. Then when you get the ground run from the house to the garage, run it to the ground bar also, and REMOVE the flattened bar in the pic that is attached with the green screw.
With only 8 breaker positions, and two of them being 240v double pole breakers, then you will never have more than six handles, so by code, you do not need a disconnect at the garage.
Understand that at the house, if it has the main disconnect in the main panel, then it will have a ground/neutral bar that may be shared, but beyond that pane, it must remain separate, ground and neutrals.
If you have a disconnect on the outside of the house at or near the meter, your house panel will have two separate bars, one for grounds only, one for neutrals only.
The ground wire you will run from the house will only need to be a #10. I hope the wire is in conduit and is THHN type wire. Code allows a #10 ground on #6 circuit wires. Depending on the size of the conduit, you could end up pulling all of the wires back out and then pulling them (with the ground) back in again. Very problematic running a fish tape thru a conduit with wires and getting it back out again.
Charles