Running underground wire to a shed
You CANNOT move the overhead line to underground. Everything from the enclosures, attachments, and the cable itself are different. Basically you'd have to remove the overhead line and associated parts and re-do the run with conduit (or direct bury). Where or not you can use the existing panels and such would need to be determined on site. 

In my strong opinion this is not something for a casual DIYer. This is only something for a someone with a good bit of experience in this type of work.

With only knowing you will be running a welder it is impossible to accurately say what wire to use. Personally, with a home shop you'd almost certainly be fine with a 90-100A feeder. You can use 100A MHF cable (mobile home feeder) which is readily available anywhere. IMO anything smaller isn't worth it in the long run. 
One thing, due to the length of the run, you may want to drop the feeder breaker to 70A to curb voltage drop. Again, this all depends on what will be used at the same time.