Wow, lot's of info to absorb!
1. Don't dig a trench until you have utility locates completed. It is a free service that utility companies will come out and flag phone lines, gas lines, cable lines, fiber optic, etc. etc. Look in the phonebook or call a local excavator to find out the phone number of who to call to do the locates. Different parts of the country have different ways of contacting the utility companies. It's required by law.
2. Dig the trench for the electrician. He will tell you to either trench to the outlet on the house or back to the main panel. Either you can do it or go to your local labor service and hire a young guy to do the dirty work for you. Plan on 18" deep minimum (tell your hired help to go 20" as they never dig as deep as you tell them to). Electricians don't like to get dirty, wet or cold. Unless you are rich, don't ask them to dig. BTW they also don't like to get permits unless absolutely neccessary, listen to their recommendations on this.
3. Hire an electrician to wire it if you are unsure of what you are doing. The wire size is based on the circuit breaker size not how far you run the wire. If you are running water heaters the electrician may want to run a separate circuit. If you only need a light and outlet he may just tap into your existing outlet depending on what's already on that circuit. A 15 amp breaker will use a 14/2 w/ ground wire, a 20 amp breaker uses a 12/2 w/ ground. The conduit is better than direct burial wire as it is harder to damage the next time you need to dig a hole and you forgot where the wire was buried. You will want to make sure the circuit is GFCI protected. This just means it is wired like a bathroom or kitchen outlet that will trip if you drop a hairdryer into the tub. You don't want to get electricuted washing down your coop.
4. If you run conduit, don't worry about a pull string yet. Just keep a shop vac handy for when the electrician is ready for it.
5. Have the electrician finish the conduit connections at both ends. Depending on the electrician, he may want to install a slip connection at the house so if you have the soil frost heave that the conduit doesn't break as the soil move in winter/summer cycles.
6. If the conduit is short the electrician will just run a fish tape through the conduit to pull the wire back through. If it is long (over 50') he may want a pull string. If so, attach the shop vac with duct tape to the end of the conduit and turn it on. Go to the other end of the conduit and start feeding pull string. It goes really easy if you attach a wad of something on the end of the string so it sucks easier. On 2" conduit, I commonly use a plastic grocery sack, on smaller conduit you can use a smaller piece of of the sack.
7. Let the electrician pull the wire and turn on the power. It should not take very long for an electrician to do the job. Ask for a price prior to them proceeding. Typically you will pay more if they just give you an hourly rate. People naturally are optimistic on how quickly and efficintly they work and usually a bid is better than if you pay just hourly.
8. Fill in the trench. Or better yet, give the electrician a beer and compliment him on his great work while the young guy fills the trench for you.
Let us know how it goes!