</font><font color="blue" class="small">( Would it make sense to run from the outlet on the outside of the house to a sub panel installed in the barn and then from a circuit in the sub panel to the fixtures in the barn? )</font>
No. Your outside plug in is wired with either 14 or 12 ga wire. These are _not_ big enough for the 150 foot run. The longer the distance, the bigger the wire needs to be. Voltage drop is the enemy.
Even for just a wimpy little 20 amp box in your shed, you need to run #6 wire to the shed.
It is not legal to do this wiring unless you start from a box (main breaker, or a load center) and end in another box (load center) in the shed, with breaker(s) there too.
Frankly, one is a fool if one doesn't run a 60 amp 240v service to a shed. You already need a thick wire, and you need a box on both ends. Might as well do it right, and pull more amps out to there, so you can plug in an air compressor or table saw or something. The 'extra' cost to you to go from 20 amp 120 to 60 amp 240v is upgrading the wire from (2) #6 to (3) #4 wires, and buying a couple larger fuses for in your basement main pannel. _Silly_ not to do so........
Now, if you are starting with a 60 amp fuse box in your house, and no room to add anything to it - I understand, try to get by with what you have...... In any case, you cannot tap into the electric outlet on your house. That will not work, you will not get enough useful power 150 feet away to plug anything in, ignoring the code violations.....
Here is a voltage drop calculator to play with, check out the difference in voltage, amps, distance, and copper vs aluminum. (Use the right grease & clamps if you use aluminum feed wire!!!!!!)
http://www.elec-toolbox.com/calculators/voltdrop.htm
--->Paul