How can I fix my garage's existing wiring
Hold on.  Why is it 10 AWG?  You're always allowed to use bigger wire than is required, but why did they do it?  It's important. 

By the way, a local breaker will not trip first.  In most cases, it's at least 50/50 that that the breaker in the house will trip first, closer to 100% if it's raining :)

Factoring for distance

For a long cable run, you need to examine your actual loads (not breaker trip rating) and the distance, and consider the voltage drop you would suffer on that circuit given those actual loads.  If you're a cable salesman, you want to keep it under 3%.  If you're a cable buyer, you want to keep it under 6%, and if there are other segments of wiring, make sure they all together don't exceed 8%.  8% isn't going to break anything. 

Most people do not realize the above, and take the cable salesman's default of breaker trip value / 3%.  That calls for a bump in wire size every 58 feet @ 120V, or every 115' @ 240V, give or take 10 feet or so. 

If your garage is >60' from the house, you need to consider your loads and recompute voltage drop for your loads and percent preference. 

One option: change the circuit to 240V

Right off the bat, doing so cuts your voltage drop in half, and your power drop by 3/4.  After this refactoring,  you could definitely fuse the circuit for 20A -- and just like 120V, 20A is the cap for "normal" branch circuits.  You would need to use 240V lighting (which is easy, many lights are 120-240-277V multi-voltage these days).  And you would need to change all your receptacles to NEMA 6, and use 240V appliances obviously.  You could use an isolation transformer for 120V appliances.  

Since this is still one circuit, you don't need to install local grounding rods.  

Another option: Multi-wire branch circuit

If the existing wire is conduit, this will be easy, but you will need to replace the 10/2 cable with individual wires.  You get to do your voltage drop calculations based on 240V if both legs are loaded at least halfway - so if you have 6A of lighting on one leg, and 12A of tool on the other, you're golden.

This is still a single circuit, so you still don't need grounding rods.  

120V Subpanel

This is the solution you are thinking of.  Without running an additional neutral, you can wire the subpanel as a 120V subpanel.  Pick a tiny panel (e.g. 18 spaces)* that has a main breaker (mandatory), and just split the 120V hot wire to both sides of the main breaker.

Technically, you don't need a main breaker only a shutoff switch, but the cheapest way to get large-amperage shutoff switches is main breakers in panels.  Also, it's morally important that the shutoff switch be readily obvious, and a larger "main breaker" set above the other breakers does the job.  Code technically allows a back-fed breaker with a label, but I think that's too confusing to someone in a panic, in a hurry, in the dark.  

Since its' a subpanel, it will need 2 ground rods set at least 6' apart - yes, in addition to the ground wire going back to the house.  It only needs one ground rod if it passes the ground impedance test, but that test is not DIYable, and ground rods are. 

120/240V subpanel

If it's in conduit, and you can run 3 wires + ground, you can make it a 120/240V subpanel, in the normal way subpanels are done. For least voltage drop, try to have loads balanced (or at least 50% balanced). Other than that, see above. 

Transformer-fed panel

Lastly I'll mention the option of using the /2 cable to feed a 240/480--120/240 transformer.  You get to keep the same cable, but you are able to deliver 120V/240V, and voltage drop on 120V loads is reduced by 3/4.

In this case you obtain a 5 KVA transformer (Craigslist sometimes has them in the $100-ish range) and feed the primary with 240V from the house. The secondary feeds the garage panel, which is a "main panel" (because it's transformer-derived) so we need ground rods again.  The transformer automagically converts your 120V draws into 240V at half the amperage, which means 1/4 the voltage drop.  You can also jumper the transformer to give you a few volts "bump" to compensate for voltage drop.