What type wire to use for 240V, 30amp, 6-30R, with 60' run?
We don't generally worry about upsizing 240V wire until about 180', so 10 AWG will be fine.
You only need 2 hots + ground.  If you use cable, consider using /3 (2 hots + neutral + ground).  That's for 2 reasons: it's round; and it improves ability to reuse the wiring for other purposes later.
You can use any of the following wiring methods:

THWN/THHN wire in conduit - you might as well use stranded wire, which will be an easy pull
Cable in physically protected locations e.g. inside walls
Cable inside conduit provided the conduit is large enough

By &quot;large enough&quot; conduit, I mean the conduit's actual inside diameter must be 138% of the cable's widest width (hence the importance of round cable).  However pulling any cable through conduit is quite hard, and upsizing just to make the pull easier helps a lot.  A novice DIYer can easily wind up in a situation where you run out of swear words, and have to call in a professional :)
Anywhere you transition between THHN and cable, that must happen in a junction box.  I recommend a 4x4 metal junction box so you have enough cubic inches. If you plan to ever use that conduit for other circuits, use THHN - multiple cables requires a HUGE conduit.