There is a pond in Houston called "Lake 288", 25-30 feet deep. Many scuba divers come here to test equipment, do some dives for experience, certification, etc.
I know that there is a buddy system that is a must when diving. However, if I go to check my equipment, stay on the shallow end, etc, do I need a buddy? With 20 feet a regular depth, not much can happen and become an emergency.
Would it be utterly foolish to go solo?
You will find the whole solo/buddy question has been discussed here many times. I don't say this to admonish you for bringing it up, but rather to point out that this is one of the most controversial subjects in recreational scuba. Many new divers are tempted to go solo because they don't know many other divers and can't get buddies to go diving with them when they want. This was the case with me. I occasionally will dive solo, but my first solo dives were in my first year of diving when I wanted to dive and didn't have a buddy. In a "do as I say, not as I do" moment, I have to say that solo diving is probably a bad idea. I know there are a lot of pros-and-cons on this subject. But at least with a buddy, there is someone who knows approximately where you are so the Coast Guard knows where to drag for your body.
In the US, there is nothing illegal about diving without a buddy, in most places. I do know a few places where local ordinances prohibit solo diving. (Such as the Edmonds Underwater Park, in WA state.) I still do make occasional, and very rare, solo dives. But the more experience I get, the less I'm inclined to dive solo. Getting older, my air consumption is not what it used to be. So, for example, when I'm diving in Roatan with a group, when I'm getting low, I'll go back toward the boat on my own and spend the last 10-15 minutes noodling around under the boat at 20-25ft by myself. I guess that's sort of solo diving.
All that said, I'd really recommend a diver to have at least 100 dives before considering diving solo. And you should look into a class. One agency, I think it's SDI, has a solo diver certification. Also, I know a enormously experienced instructor-certifier who used to teach a "diving self sufficiency" class.
While I agree that a 20ft dive is less hazardous than a much deeper dive, don't forget that you only have to hold your breath for 4-5 feet to kill yourself, or injure yourself badly. If nothing else, take a redundant air supply like a pony bottle, a spare mask and at least 2 knives.