What's a good method for beginning to set the action on a newly strung guitar?

It really depends on the type of bridge/trem used.

That was a nice, thorough answer, and I'm sure it helped the person asking the question.

While it does depend on what type of bridge you have, setting action is very simple. First, you need to make sure that the guitar is strung to pitch. Tune it to whatever tuning the owner uses. Next is adjust the truss rod if needed. You want a very slight amount of relief in the neck. If the neck is perfectly straight or has backbow in it, the strings will possibly touch the frets and cause buzz. Remember, just a slight amount of relief. Next, retune to pitch if adjustments were made. After that, the nut height needs to be checked. Nut height is checked by fretting each string on the SECOND fret. There should be enough clearance between the FIRST fret and the string to fit a sheet of paper. If you have too much clearance, you will need to file the nut slots. This should not be done by anyone but a professional. If you dont have proper nut slotting files and experience, bring it to a shop, and just pay for a professional setup. Anyway, if your nut slots are too low, you need a new nut. If the nut slots are the right height, you can now set your action. A general starting point for action height is 2/32 on the high E and 3/32 on the low E. Action is measured at the 12th fret, and the measurement is from the bottom of the string to the top of the fret. Bridges that are pre- radiused such as Floyd's and Tune-o-matics are the easiest, all you do is raise/lower the outside posts. Bridges with individual saddle adjustments can be a little difficult. You not only have to adjust the action of each string, but you must also maintain the radius as well. I hope this helps.
Bob

It also helps when the original poster replies with an answer

In most cases the nut will not need any work unless it was replaced and cut wrong.