You need a new bike
You should probably take a day or two off the bike to let your hands/wrist recover. You might also take a look at your bike fit. Hand and wrist problems are generally caused by too much pressure on the handlebars. You may need to raise your bars up a bit. Most bikes come with pretty low bars relative to the saddle height. For new riders, this can cause various problems, most commonly numbness. You might also pay attention to your hand position on the bars. If you ride with your wrists bent back, you can irritate the tendons.
Yup, you may be overdoing it and you may be putting too much pressure on your hands and wrists. Try researching bike position/fit with google. There's a lot of science and opinion out there and one has to experiment. I'm still fiddling after years of riding on my road bike as I extend my distance. Good riding position is a funtion of experience, physiology, distance of rides, frequency of rides and to a much lesser degree age.
My mountain bike position is locked-in. It's less critical as you move around a lot.
Scottish Canuck in the US
This is common for many new riders. As others have indicated, make sure your bike fit is dialed in. Check with your LBS.
Use the search function on this forum and you will get some good advice on rider position.
Most important advice here once you get back on the bike is to relax. Keep a slight bend in your elbows and don’t hold the grips too tight. Concentrate on relaxing your upper body. Shrug your shoulders every 10 minutes as a reminder to keep your upper body relaxed. If your upper body is tense and your elbows are locked, your wrists and forearms are absorbing all the road shock and this will lead to stress/pain.
Don't have a death-grip on the bars!!! If you can't ride with all your fingers loose and wiggling, you're not placing your hands on the bars right... The fingers should be rotated outwards a little so that the meaty outside heel of the palm is resting with all your weigh on the bars. You shouldn't need to have the fingers gripping anything...