Killarney Hostel

I therefore intend to make the Ring of Kerry, Dingle Peninsula tour and visit Mizen Head, while staying at a hostel in Cork. Would it be better to just stay in Cork or should I head to Killarney for 2 nights? How long does it take to get to Galway from Cork to Killarney and Galway?

It's quicker to get to Cork from Galway, but there is nothing wrong with staying in Killarney for two nights!
Distances:
Depends where you're staying in Galway .... City:

Galway City - Cork 201.3 km ... 3 to 4 m [could be shorter depending on where you are]

Galway City Killarney 213.56-km ... 3 h 14 m

Good luck!

I think it will be a good idea to stay in Killarney for 2 / 3 nights and see the Ring of Kerry and make GALWAY it will give you the best two places to enjoy
I lived in Cork for 16 years and enjoyed every day of my life and I used to go to Kerry and a lot lot of time with family and friends
Enjoy the best of Ireland

Apart from the bottleneck in Charleville, Cork, Galway is a very fast track.

Cork, Galway, because itll be way easire and less Expess. DUH ... come on common sense Huns