Good Evening All,
Over a year ago, I had an electrician come out and install a sub panel in my new garage. At the time,
he installed two 30 amp breakers linked together that feed the garage.
I
DO hope that you really mean a single dual-pole breaker, as opposed to two single-pole breakers lashed together with a handle tie. Either way...
My question- If I wanted to add more outlets to the garage, how big an amp breaker can I use for this circuit?
It's a real shame that you spent all that money, and (apparently) only got 30-Amp service to the sub-panel. It would have been only very marginally more expensive to upgrade that to at least 50-60 Amps at that time; then you would have had plenty of spare capacity to add additional branch circuits in the garage as the need arose. But to upgrade it now would near-certainly require that the feeder cable from the main panel to the sub-panel be replaced with heavier wire, which will likely be neither easy to DIY nor cheap to have done by a pro. You cannot (at least usually) simply replace that breaker with a higher-rated one, as that would lead to the feeder no longer being adequately protected.
That said, there just MIGHT be an exception here. The electrician who installed your sub-panel MAY have undersized that breaker, relative to the feeder line it is intended to protect. There isn't really a GOOD reason for him to have done this; but I've seen it happen before. Maybe all he had on his truck that day were 30A breakers, and he figured that would be "good enough" for the lightweight stuff you planned to do in the garage back at that time. (That would be schlock work; but again, I've seen it happen.)
The only way to know for sure is to determine (and report here)
EXACTLY what type/size wire he used for that feeder line. It's outer jacket should be marked every foot or so with, among other things, the AWG of the conductors; and since that wire is only a year old, those markings should still be quite readable. Hopefully, the cable will be accessible/visible somewhere along it's run, so you don't have to pull off the panel faces to check the markings. But if you do have to open up one (or both) of the panels, be
VERY careful -- there's a lot of 240V "stuff" in there, now with exposed metal connections; don't be poking your fingers (or Ghod forbid, a metal screwdriver) in there!
Also, approximately how long (as the wire flies) is the run between the two panels?
I'm thinking a 15 or 20 amp max because that will be maxing out the load? Am I wrong?

HUH?!?
If you're talking about the breaker protecting the feeder line, then per your opening paragraph, it's already 30A. Going to 15A or 20A would be a DOWNgrade.
If you mean how large should a new branch circuit (sourced at the sub-panel) to support those new 120V outlets be, the standard answer is "20 Amps" (and BTW, it probably ought to be on the phase opposite to whichever phase the existing 20A breaker feeds from). As others have pointed out, the total amperage ratings of all your branch circuits put together CAN (and often does) exceed the rating of the "Main" breaker (or, in a sub-panel, such as in this case, the breaker feeding that sub-panel). What matters is the total draw of all the actual loads which will be in simultaneous use. Even so, 30A isn't much.
I'm not using anything very strong in here. The compressor and coffee pot are the two things that draw the most juice.
Compressors (and their electrical requirements vary WILDLY. Without knowing what sort of compressor you have, it is impossible to even guess at what it might draw. And FWIW, coffeeposts can draw more than you might think.