If I told you how many cuts, scrapes and pokes I've gotten on my hands! From gardening, to being the family chef to just doing things around the house, hundreds. They heal like any other wound on your body. There is no extra danger from having a small wound on your finger than on your leg, back or face. No big deal. Put some Neosporin on it and call it a day. It will heal. good luck
Probably not a big deal, but I got a finger infection from a manicure once and was really worried also. It swelled up and was pretty sore. I soaked it several times a day in warm salt water, and it took the infection away.
Puncture wounds don't tend to bleed to badly unless you hit some type of larger vessel. There are two worries you should have with this injury: tetanus and infection.
If the wood stick was outside or had exposure to dirt, and you haven't had a tetanus booster in the past 10 years, I would go get one. It's a VERY slim possibility, but a booster shot will then cover you for 5-10 years in the future anyway. No loss.
If the wound starts to get red, hot, swollen, more painful to the touch, or the joint gets hard to move, go see a doctor. An infection is very treatable, but if not observed and it worsens you don't want it to start involving that knuckle joint. No sir!
All in all, you're probably OK. Wash the area, bleed it a bit if you have to, and then put neosporin on it with a bandage.