What size breaker should I use in the main panel for a sub-panel with four 15A breakers?
If you are running a proper sub-panel on the North American model (i.e. 240V + neutral and ground) a 30A double-pole breaker and 10Ga wire will permit 4 15A 120V circuits to run at full capacity.

If you are using existing wire, that sets the upper limit of what you can feed it with, whether or not that is adequate for your needs. Else, you will be replacing the wire to meet your needs.

If the panel might be expanded later (i.e. 4 circuits now, but with space for additional circuits) and you use larger wire, you can feed it up to the ampacity of the feed wire and/or sub-panel rating.

If you are running a single-phase (120V) feed you will probably not be happy, and you would need large wire to feed it anywhere near full capacity - which is why the normal/"proper" method is a 240V feed.