That link from Dark Hippo is great for wrist flexibility and strength. I would highly recommend those movements as well. Personally, I started experiencing wrist pain a few years ago when I first started calisthenics as well. Today I experience little to no wrist pain after implementing several things detailed at the bottom.
I believe my pain stemmed from Carpal Tunnel Syndrome, so this answer only focuses on that particular wrist pain.
Carpal tunnel syndrome (CTS) occurs when the median nerve, which runs from the forearm into the palm of the hand, becomes pressed or squeezed at the wrist.
The NIH lists these as treatments for CTS.
- Splinting. Initial treatment is usually a splint worn at night.
- Avoiding daytime activities that may provoke symptoms. Some people with slight discomfort may wish to take frequent breaks from
tasks, to rest the hand. If the wrist is red, warm and swollen,
applying cool packs can help.
- Over-the-counter drugs. In special circumstances, various medications can ease the pain and swelling associated with carpal
tunnel syndrome. Nonsteroidal anti-inflammatory drugs (NSAIDs), such
as aspirin, ibuprofen, and other nonprescription pain relievers, may
provide some short-term relief from discomfort but haven’t been shown
to treat CTS.
- Prescription medicines. Corticosteroids (such as prednisone) or the drug lidocaine can be injected directly into the wrist or taken
by mouth (in the case of prednisone) to relieve pressure on the median
nerve in people with mild or intermittent symptoms. (Caution:
individuals with diabetes and those who may be predisposed to diabetes
should note that prolonged use of corticosteroids can make it
difficult to regulate insulin levels.)
- Alternative therapies. Acupuncture and chiropractic care have benefited some individuals but their effectiveness remains
unproved. An exception is yoga, which has been shown to reduce pain
and improve grip strength among those with CTS.
- Surgery. Carpal tunnel release is one of the most common surgical procedures in the United States. Generally, surgery involves severing a ligament around the wrist to reduce pressure on the median nerve. Surgery is usually done under local or regional anesthesia (involving some sedation) and does not require an overnight hospital stay. Many people require surgery on both hands. While all carpal tunnel surgery involves cutting the ligament to relieve the pressure on the nerve, there are two different methods used by surgeons to accomplish this.
My Own Experience
The first thing I tried was to simply get my wrists into a neutral position in my day to day activities and for handstand related movements. First, I built myself some parallettes, so I could do movements with a neutral grip. This helped quite a bit while I built up strength in my wrists, but it didn't fix it.
Then I bought a split ergonomic keyboard and a vertical mouse to use at work. I also completely removed video games from my life. This helped a bit, but it still wasn't enough.
I finally tried sleeping with wrist braces on, and my wrist pain went down instantly. It turns out the main cause of my wrist pain was sleeping with bent wrists. Within a few weeks, there was virtually no pain/numbness. When I don't sleep with wrist braces, I feel it in the morning.
Eventually I progressed off of my parallettes as I got stronger, and my wrists stated feeling numb after long workouts again. I bought some wrist wraps to use during planches and handstands, and now the numbness is gone again even after long workouts.
From personal experience, I would recommend buying a set of wrist braces to sleep in and some wrist wraps for workouts. These two thing helped far more than anything else and are relatively cheep solutions.