Sub Panel feed to existing garage
You might as well go all the way with your subpanel upgrade
There is absolutely no reason to put a dinky 10-space panel in your shed when you could get a 24-space or 30-space, 100 or 125A, main breaker panel for a minimal additional cost (the price differential we're looking at here pays for oh, a pizza or a few fancy cups of coffee, depending on where you are).  Keep in mind that the main breaker in your subpanel is simply a shutoff means for the shed, so it being larger than the feeder breaker in the main panel is not an issue.  Don't forget to buy and install the correct grounding bar for your panel if it doesn't come with one pre-installed!
There are better ways to invest in copper
You're far better off pulling a 2-2-4-6 Al mobile home feeder cable for your run than doing anything with copper.  It's one-third the price, has 50% more current-carrying capacity (90A at 75°C), and will happily fit in 1¼&quot; Schedule 80 PVC conduit.
While you're at it, you might want to use some of that money you saved by not running copper to lay a continuous run of PVC conduit belowground.  While 1¼&quot; will work for this, going up to 1½&quot; gives you room for future upgrades, up to 100 or 125A even depending on the panel you get.
At the main panel, you'll need an 90A breaker of the correct type for your panel (although you can use a 60A breaker if you already have it fitted) to take full advantage of the mobile home feeder cable.
As to that grounding situation...
You are correct that you'll need ground rods at the shed in addition to the grounding wire in your feeder cable.  This is handled with some bare 6AWG copper run from the garage subpanel's grounding bar to two 8' ground rods driven 6-8' apart, as a single rod might not get a low enough resistance to earth to protect your shed adequately against wayward natural electricity.  Don't forget to pull the green bonding screw or strap out of the subpanel while you're doing this installation, for that matter!
TORQUE ALL LUGS TO SPEC
Last but not least, you'll want to use an inch-pound torque wrench or torque screwdriver to tighten all connection (breaker, panel) lug screws to the rated tightening torques marked on the breaker or panel labels.  This is required in 2017 NEC 110.14(D), and is a good idea even if you're on a prior edition of the NEC, lest your electrical work give your garage the loose lugnut!