I have yet to lube my chain (my first ride on the bike was the first day I rode into work). What type of chain lube should I use, and how often should I use it? Should I clean it on a regular basis?

Fairly regularly, yes- it's been a while since I rode a derailleur bike, but every couple hundred kilometres should be fine. I just turns the cranks backwards and wipe the chain down with a rag. I might or might not wet it with a cleaner first. I tried chain cleaning devices but found them to be a messy time-consuming pain that didn't necessarily give me better results. For lube, I've used Finish Line, Campbell-Hausfeld air compressor oil, and chainsaw bar oil They all work well. Basically anything that's actually meant to be used as a lubricant, i.e., not WD-40, since it's a solvent.

What other type of daily/routine maintenance should I perform on my bike?

Squeeze the tires every every few days or so to check air pressure. Be aware of wear and tear on consumable parts: tire wear, brake pad wear, cable stretch, stuff like that.

Riding on drop bars is a new concept to me, so riding with my arms fully extended has made the inside of my elbow pretty sore. Is this common and what should I do about it? Stretch, lift, ice, etc.?

It's not common. Icing your elbow may help a bit, but it's a band-aid solution at best. Pain is a signal that something doesn't fit right. If you're riding in the drops, try riding the hoods; if the pain persists, you may have a problem with the way your bike fits you. Without seeing you on it, I'm guessing it's a bit big (too long a top tube, and thus too great a reach), which can possibly be mitigated by getting a shorter stem. Your LBS can give you advice about this.

I live in an apartment with ample outdoor storage to lock the bike up in. I would like to get a small bike maintenance kit together (lube, rags, allen wrenches, etc.). Do any commuters on here have a good no-frills kit list that they'd like to share?

I keep a bunch of rags at home (I'm partial to old socks, I can stick my hand inside and grab the chain to wipe it down without getting my hand dirty; old t-shirts are good for flossing between cogs). Allen wrenches are good, a pump is a must, as is a patch kit. For routine, day-to-day stuff you don't need much more. If you plan to do repairs yourself, the a bigger toolkit is handy: spoke wrench, chain tool, third hand (for holding the brake pads against the rim while you make adjustments, etc- so handy!).

I realize I could search through all the forums to find varied answers to these questions; I just thought I'd drop them all in once place and see what you folks could come up with. Thanks for any insight!

Parktools.com has a fairly extensive repair help section that's pretty useful. The search function here leaves something to the imagination, but there's eleventy-thousand of us around anyways, so ask away