Hello I was doing some remodeling in my kitchen and found a 2 pole 30amp 10-3 wire abandoned in a junction box under my kitchen island. its wired to my panel but the breaker was turned off and not labeled. I checked and it is live when the breaker is turned on. My question is can I branch out from this with smaller guage wire e.g. 12-2 or 12-3 for outlets and a sink with a disposal for my new kitchen island? I'm thinking no but not quite sure.
As long as the existing cable has a grounding conductor what you propose is absolutely fine, although you must replace the breaker with the proper two-pole 20A breaker. Although not required I would definitely place a label on the wire in the panel stating it is to be used for a 20A circuit maximum.
I would place the 10/3 in a box and branch off with two 12/2's going to your new loads. This will give you the two 20A dedicated circuits you are looking for.
You could put in a sub-panel (which can be greater than or equal to 30 Amps capacity; the load to it is ruled by the wire and the breaker in the main panel feeding it, not "what size it says it is") and breakers to feed branch circuits, if you need (or have use for) the full 30 amps.
30A 240V Dual === 40, 50 or 60 A Sub --- perhaps 4 120V singles - 20A for all would be fine in many cases (with a small possibility of tripping the 30A Dual if all were loaded fully at the same time) or they could all be 15A if being cautious (but not if they feed countertop receptacles, that require 20A) or you could use 2x 20 and 2x 15...