Poplar Tree Lake and Piersol Lake in Meeman-Shelby Forest are decent options. Fish light tackle for bluegill/bass.
Look up Lyle's Lake near the Shelby Forest General Store in north Shelby County. I think it costs $5 to fish there. I don't remember the days that they're open, but they have some jon boats that you can use - you have to bring your own boat gear though (life vest, paddle, and a trolling motor/battery if you want one - the are nice to have!).
There are several ponds around Shelby Farms (a giant urban park in East Memphis) - use Google Earth/Maps to find the ones that are out-of-the-way, and go exploring. There are a couple of big lakes in there, including the main one that was recently constructed. It looks rather desolate to me, so I haven't fished it in many years. If you have a bicycle, that can widen your options, as some of the ponds are not that close to the road. This is probably your most interesting option, as there are quite a few small bodies of water around there and you can have a bit of adventure...I did this a lot back in high school. They are best for bream, bass, and sometimes crappie. I've never caught any eater catfish, only bullheads (aka "mud cats").
North Mississippi and East Arkansas have some waterways that open up your options a bit. I don't have a lot of experience with them, but they are certainly options. Also, you are within about a 3-hour drive of some great trout fishing in Arkansas (Little Red River and Spring River), so plan on doing that sometime during the heat of the summer. That cold water is nice
The best bank fishing spot in the entire western third of the state is (in my humble opinion) the tailwater below Pickwick Dam, within 150 yards of the dam wall. PIckwick is the nearest Tennessee River impoundment to Memphis.
From my house in Bartlett (NE Memphis suburb) that's a 2-hour drive. Yes, that sucks. When I was younger and single or with no kids at home, I'd make the drive several times in the late spring, and a couple more in the early summer. Nowadays, I might only go twice a year. You can catch 10 species of fish in a single trip if using a variety of baits, so I really enjoy the variety in provides.
I'd also encourage you to save up about 1,000 bucks and buy yourself a jon boat and trolling motor down the road. Even with no gas motor, there are several lakes in the area that you could get into easily and it would widen your options quite a bit. Another option would be a kayak, if you're into that kind of thing.