What do I need to know when burying some wire?

When I got a permit to run 220V underground, I was required to go at least 18" deep.

I would follow that recommendation for your situation as well. In addition to the pull wire, I would pull several extra network (copper and fiber) and coax wires. It's a lot easier to pull all the wires in one bundle than to pull them one at a time. Make sure the wires are all water proof.

After filling the trench about 80%, put a plastic tape in the trench. The tape is bright colored and contains writing along the lines of 'buried cable'. Then finish backfilling.

Document the location of the trench and of course the contents of the conduit.

Call the locator service to make sure you don't cut other wires/pipes when digging.

It against wiring code to pull "high voltage" wires (110V or 220V) along with "low voltage" wires.

Be smart, call your local utility providers.

http://www.energyindustryphotos.com/pipeline_blowout_photos_and_natu.htm

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When installing data lines between buildings, ground potential can be a serious problem. You could end up with a lot of burned up equipment if connected incorrectly.

I would suggest using optical isolation (fiber optic NIC or adapters), point-to-point wireless or hiring a professional.

http://cim.pennnet.com/articles/article_display.cfm?article_id=62689

You didn't specify either way, but if you're running power, you really need to talk to your city or county inspectors. They're typically very helpful to people who are trying to do things the right way. They'd offer suggestions for low-voltage (data) lines as well.

Make sure you check with your local utilities to ensure you'r not going to cut into a 10,000 volt power line or crack open a sewage pipe.

Hire a contractor. He'll know everything and can help will pulling permits.

You should be using underground electrical conduit.
To answer your question about how big, it depends. On how many cables, of what diameter etc. Make sure you stay within 300 feet of total run length, or you won't pass a LAN cable test
Here is a good site that shows you how to calculate how big to make your conduit. Make sure you leave room for expansion, bad wire replacement etc. AND a pull wire!

Also you may want to consider if you're living in a colder region that the cables are deep enough that they will be below the frost line.

Make sure you've checked with the local authorities on the subject so that you don't violate any regulations and/or get yourself killed.

But really if i were you i'd outsource that unless you've got some experience with it. Supervise if you can to get the experience, or hire a contractor.