Gauge, amperage, and breaker box
If your ambition for the shed would call for more power when you can afford to fix things, put in a sub-panel with that in mind, not one limited by your 12Gauge (20Amps - depending on the exact wire you have, 120V or remotely possibly 240V, 2x120V) feed wire.
Assuming a 12 Ga 120V feed (white, black, bare or green) the breaker at the house should be 20A and GFCI (because I'm assuming the burial may be sketchy, among other reasons) and that is actually adequate protection for the whole circuit as it stands, IF that's how it's fed.
i.e. you can certainly put in a sub-panel with an eye to the future, but until you upgrade the feed to the shed, you don't actually need to.
General advice here is to go big on sub-panels. Among other things, small ones are not that much cheaper than big ones, and the &quot;size&quot; does NOT need to &quot;match&quot; the breaker feeding it, while having plenty of &quot;spaces&quot; to put breakers in is very useful, and tends to scale with &quot;size&quot; - so a 100 or 200A panel with 12-24-40 &quot;spaces&quot; might be perfectly reasonable even if you were only planning to run 50A to the shed. The &quot;main breaker&quot; in the subpanel will never trip in that case, but it DOES serve the useful purpose of being a local shutoff for the shed.
You may be perfectly happy with a smaller panel, given that your current ambitions are 4 receptacles and 2 (light?) switches, which would only be 6 circuits at most and perhaps as few as 2, but look at your options, costs, and any potential growth before going tiny.
For now, based on your wire, you only have 20A available, most likely 20A at 120V.
That's a way to measure the wire size, but the wire size should be imprinted on the jacket of the cable (or printed on individual wires that are not in cable, which does not appear to be what you are describing. If your conduit extends all the way, or you dig up the cable to put in one that does, that's what you might replace the cable with.) So you should be able to determine wire size just by reading the outer sheath.