This is a topic that is near and dear to my heart. Yes, you can spin a sock yarn on a spindle. That said -- spinning on spindles to sell yarn is probably not very efficient. Peaceful, relaxing, but not efficient.
I've been working on my wheel-spinning for sock yarn lately and that is eye-opening in and of itself -- wheel ratios are becoming much more familiar to me vs. my earlier, DK-to-worsted spinning on wheels. Hooray for 24-to-1, I finally got my Better Roberts Accelerator working. But wheels aside -- congratulations!
Here are some good links on spinning for socks:
I would note they are at opposite ends of the spectrum -- spinning firm vs. spinning loosely. I think the biggest thing to consider may be (a) the wear pattern of the sock (b) the shoe it's going into. For a tight fit, you want a fine yarn that fits in a dress shoe -- something knittable on a size 0 or 1 (US) needle; but you can knit great boot socks on a size US 5 needle with worsted weight yarn, too.
Besides a WPI tool like the one from Nancy's Knit Knacks, I find the WPI Control Card from VIP Fibers to be a handy gadget (if only I could keep track of my gadgets ...) but remember, your yarn may bloom and fluff up a bit after it's spun and plied ... so sampling may be useful early on in your learning.
I'd probably hand a beginner a Romney wool, maybe a Shetland. But that depends -- if they're a sock-yarn beginner but know how to spin, a superwash Merino's not out of the question, and there's a yummy Merino/Tencel blend in my stash to be sock yarn. My current sock yarn blends on bobbins/spindles include: superwash merino from Crown Mountain Farms, a blend of superwash/romney/silk (from Abby!), and a CVM/Viscose/silk noil blend by Crosspatch Creations.
Having Mohair, nylon, viscose (aka rayon) and/or silk in blend with wool can add strength, so most spinners I've talked with recommend blends generally, for socks.
(posted by me this day on LJ:spinningfiber)
The Bellwether sells a mini sock spinning kit which includes fiber, a pamphlet on spinning for socks, a mini sock blocker, and a pattern to knit your mini sock.