I am running underground wire to barn approx. 450 feet. Will have 3 light fixtures & 4 recepticals. What size wire is recommended ?

Are you planning a sub panel? Your question can't be answered as posted since we have no idea what you plan to plug into the recepticals. What will be the load? Small drill or a welder? -makes a big difference! 120 and 240?

Are you planning a sub panel? Your question can't be answered as posted since we have no idea what you plan to plug into the recepticals. What will be the load? Small drill or a welder? -makes a big difference! 120 and 240?

I was not planning a sub panel, do you recommend installing one ? The recepticals are more for convinence. During the winter months there will be two water tubs that draw 100 watts each to keep the ice off the water. I was planning to run 12/2 wire from a 20 amp breaker in the house. If I used a sub panel what size wire would I run from the house? And what size wire from the subpanel.

You are going to have a very hard time with a 12 @120. Voltage drop is 18%, way too high. Even #8 is borderline @120VAC. You need to consider running 240 volts to a small subpanel, double 30 Amp at the house, 10/4 UF to the subpanel and then 12 from there. Even that is probably good for just 10-15 Amps.

Voltage Drop Calculator (http://www.calculator.net/voltage-drop-calculator.html) is a good tool to show you exactly what you need.