Outdoor 50 amp 4-wire Outlet Install
You're on the right track

Since you have a suitable panel, and enough spare space in it, I would go for what you describe, using 3 6AWG wires for the 50A/240V circuit and 2 12AWG wires for the 120V circuit (as well as a 10AWG ground, if you're in plastic conduit that is).  You'll want to use a GFCI breaker at the panel for the outdoor 120V circuit, by the way, as that keeps it safe from the weather, vs a GFCI receptacle outside which can get affected by water that leaks into the receptacle box.

If you're going to do this using cables, then you can use 8/3 on a 40A circuit or 6/3 on a 50A circuit for the 240V receptacle, with a 12/2 extending into the 240V receptacle's box and spliced there to some 12/2 UF or two 12AWG THWNs in conduit to connect to the 120V receptacle.