New sub panel wire sizes
For a 30A breaker, honestly, you're fine at #10 wire.  At that distance you don't need a wire size bump even for 120V loads. 

You said the word "cable" which means several individual wires in a sheath. That's not what you want, it's very difficult to pull through a conduit.  Get individual wires, of type THHN (typically dual-labeled THWN-2). They will easily fit in 1" conduit.