I've recently started biking serious amounts (to me, anyways, 12 miles or so ) and I've noticed a pain in my right hand, near the center of where the wrist connects to the palm. It hurts when pressure is applied, and sometimes otherwise. Could this be from biking? My first thought is carpal tunnel, because I also work on computers for a living, but I'd really like to avoid having carpal tunnel.
This is most likely because you are putting your weight right on the median nerve (see first picture). Padded gloves may help a bit, but I suspect your hand position is more likely to be the problem.
Other common hand nerve problems with riding can compression of the ulnar nerve, which will present itself as a different area of sensation than what you are experiencing (second picture)
If you are using road bars
I remember you were riding a road bike in the Legs are very weak question.
Next time you ride, watch how you place your hands on the handle bars. If you are using road bars, beginners often place their hands right on the bend between the Tops and the Ramps (see second picture). This would put pressure directly on the median nerve. Try riding on the tops, ramps, hooks, and drops instead of the bend between the tops and ramps. It is also good to rotate through the four hand positions rather than using just one position. You may also find that adjusting your cockpit dimensions can make some of all these positions more accessible and comfortable.
There are a couple of things you can try before throwing down serious money on a professional fit or a new bike (side note, if you buy a new bike from a shop be sure to negotiate a free fitting session if it doesn't come with one automatically).
First, the most obvious thing to do is to get some proper bicycle specific gloves with beefy gel pads on the palms. If full finger gloves are two warm for you go with the fingerless type. Seriously though don't skimp on the gloves.
The second option which is a little more experimental in nature is to lower your seat a little bit. This will take a little bit of pressure off of your hands, maybe even enough to solve your problem completely. But you have to be careful with this because lowering your seat can put strain on your knees, and your knees are the last thing you want to injure from improper fit. If your knees bother you at all, abort mission and raise your seat.
If you're still having issues after that, it might be time to look at a proper fitting bike.