I would like to run a 50amp circuit or sub-panel 400 ft to a barn/pottery studio which could supply both 220 and 110v power, to be used to run lights, heat and kilns. What size direct burial wire would I need?

That's a pretty long run... you would need #1 wire to minimize voltage drop at that distance. You will also need a ground wire... and a grounding rod at the location. How many circuits do you plan on running?

I have a 50amp receptical box w 1-20amp breaker 240v and 1-10amp breaker 120v that should be more than sufficient for my needs, so just one circuit connecting that.

Mike,

What do you plan on using the 10 Amp breaker for?

Also, I believe that you would need 1/0 AL to overcome the voltage drop caused by the distance.

OK, so now there are two different answers to the posters question.

Which is it, #1 or #1/0?

Aluminum or copper?

What amperage is being used to calculate voltage drop?

What voltage?

These voltage drop questions not only need accurate answers, but should contain the values used to calculate.

This is for a pottery shed. I have done a LOT of wiring for pottery people. One thing I know is that they are never satisfied with their kilns. They tend to go bigger and bigger.

To say you have a very remote (400' away) pottery shed that only needs a 50A feeder, for everything, makes me very skeptical.
I have had to provide 100A circuits for kilns in a residence more than once.

#1 copper, 50 amp, 240v

<Left out information>
I chose AL because of the distance/cost. Using 60c UF cable, continious load I came up with 1/0 AL

Also used 50 Amp 120/240. I calculated using the 120 to neutral and came up with the 1/0.

And you get to pick the correct route taken.