Two things to help. !) Buy a proper pair of padded cycling gloves. Not expensive but worth every cent. 2) Check with your local bike store and have a proper bike fit. It might cost a couple of dollars but definitely worth it. Usually good bike shops throw a bike fit in free when you purchase a bike from them.
I have a dodgy right wrist with plates and screws from an old skateboarding accident and I find when I ride without the gloves it can hurt after an hour or so. With gloves - no issues at all.

I've always enjoyed casual biking, though sometimes my bike (a Gary Fisher model I bought new in 1999) will sit unused in my garage for years at a time. I've gotten back into it though, since joining MFP, and now really enjoy going for 9-14 mile rides two or three times a week. (Especially since I recently discovered that just 2 miles from my house, the subdivisions end and it's just horse and cow country. Beautiful riding!)

However, I find that after about 20 minutes, my hands get a little numb. My palms, basically. It's not that big of a deal -- I shift my weight a bit, let the blood flow back in. But is there something I'm doing wrong, or could/should be doing to prevent this in the first place? Or is this just something that happens to most cyclists? I started wearing workout gloves, which have a little padding on the palm (not much), but they haven't really made difference (though perhaps it's a little better).

The issue is bike fit. Your hands shouldn't be numb in 20 minutes. You're putting too much weight on your hands and/or the angle of your hands is incorrect. You've already discovered the cause and solution: When you shift your weight back (remove the weight from your hands) blood flows back to your hands. The numbness goes away. It's a combination of blood flow and ulnar nerve compression in your wrist. This leads to bike fit and, to a lesser extent, equipment.

Raise your handle bars. This can be done with a new stem ($10-$20) or adjusting the existing stem (if it permits such an adjustment). A bike shop can help you do this (free if you buy their stem sometimes, or a small fee to flip your stem) and it's a lot cheaper than a full proper bike fitting ($250+ probably). You can also buy a different set of handlebars ($35+ maybe) that has a rise to it. I'd go with the stem adjustments first.

Keep in mind that sitting upright too far can create a different host of issues. Sitting upright shifts the weight to your backside. Don't be surprised that once you solve the issues with your hands going numb at 20 miles your butt hurts at XX miles. It's a balancing act. The solution resides within a combination of bike fit (gear/equipment), core strength (abs), and technique (changing hand positions).

The bike fit is where you should start. If this is an MTB then look at a good set of ergonomic handlebar grips (Specialized BG Contours). Also consider adding bar ends. Gloves don't help much with the numbness. However, you may as well get a decent gel set since you should have a set for protection anyhow. The gel dampens vibration a tiny bit.

Numbness is common for new/inexperienced cyclists. Chronic numbness won't go away on its own. Once you have a grip on things (pun intended) then you will know what changes you need to make.