Hi C.,
First, the "full disclosure" so you'll know my biases--I am a musician who teaches private oboe lessons and I have also taught music in parochial schools. I substitute teach (all subjects) now.
I would definitely go with a person teaching your son, not a CD, for the reasons some have mentioned--the immediate feedback, personal attention, music-reading skills, etc. And you can't ask questions of the CD, but your teacher will usually be more than happy to respond!
As some of the postings have recommended, ask around to people like the school music/band/choir teacher, church organist, any musician you know, parents of kids who study guitar or other instruments (those teachers may know someone, it's a pretty small world). If you come up dry on all of that (unlikely, IMHO) then go to a place that just gives lessons so you are not being pressured to buy anything. Is there an Arts Center/ Arts Council in your town? Might be a good place for referrals. Or alternatively you could go to the music store but leave checkbook, credit cards etc at home.
I would not recommend the "big-box retailer" instruments based on colleagues' experiences with the band instruments sold there. They do not hold up well to normal use. Once you have a teacher let him/her guide you to an instrument. Consider renting for awhile if you think this might be just a passing fancy on your son's part. Some music stores rent instruments and most or all of the rental payments eventually apply toward purchase price. Also consider that if you get a decent quality instrument, and your son quits, you can more easily re-sell it.
As far as being cheap(I am too), you could perhaps barter some skill for the lessons--cleaning house, running errands, child care, baking bread, etc. if the teacher is amenable to that. But I am not sure I would start out that way. Remember that the teacher has taken many years to attain that level of skill, and usually has to put in some (unpaid) practice time in to maintain it.
You could also consider having your son contribute in some way to the lessons' cost. I remember one oboe student of mine some years ago (when I only charged $7 per lesson) whose parents made a deal with her. (Granted, Ellen was in junior high and had more money than an 8 yo would). Ellen could have one "day off" from practicing per week. All the other days she was required to practice 30 minutes. If she failed to practice one of those days, she had to pay $1 toward her next lesson; miss 2 days, pay $2 etc. She was responsible for buying her reeds from her own money (and as I recall was VERY careful with them so they would last a long time!). Perhaps you could devise extra chores or something for your son; it wouldn't get you any more cash, but might make him take it a little more seriously.
If you live near Ashtabula Ohio I can recommend a good guitar teacher, very patient with my kid (who eventually dropped guitar lessons in favor of recorder, then cornet.....) Email me for more info.
Sorry for the long post, hope it helps!
K. Z.