I need to run electricity to a shed that is approximately 65' away from my subpanel. I assume that I can use underground wire instead of having to use conduit. I will only be supplying electricity to a small refrigerator and a couple of fluorescent lights. What wire size should be used and if I called an electrician in to do this task for me what would be a round about estimate of the overall cost?

Well, considering we know absolutely nothing about the details of this job it is impossible to even guess at a price someone would charge.
Even your location is a big factor.

For what you need a multi-wire circuit would be fine. You can run 12/3UF cable and get two 20A circuits from that.

I'd personally go with #8, voltage drop is 1.7% or 2 volts.

#12 is 4.2% or 5.1 Volts

Once you get to the shed, you can switch to #12.

The troublesome part is trenching. The wire must be buried at least 18" deep.

KISS,
#8?? For 65'??
What are you basing this on? :confused:

I don't care WHAT the load is, at 65' VD is NOT even a consideration. :cool:
Especially for such transient loads.