You can cheat on a PS2 with PS1 cheat software, as long as it's something disc-based, but I don't really recommend it. Those devices weren't that great. If it's all you've got, you can find those on e-bay. I'd just advise caution. People can mislabel, and the discs very often look alike, so make sure it's actually PS1 software before you buy. Either ask the seller, or look at the photos of the packaging and disc and see if you can find "for use on Playstation" or something similar written on it.
For the PS2 games, I think the only cheat device still being sold by its manufacturer is the AR MAX. Although, maybe new Xploders are still being made in Europe. Other than that, it's back to e-bay to find a GameShark or a CodeBreaker. I'd recommend against the GameShark. The early versions were OK, but limited, and they eventually became buggy messes with obnoxious interfaces. I have a lengthy set of the pros and cons of each PS2 device posted around here somewhere.
To cheat on all the games, you'll have to buy at least two pieces of software. However, I very much recommend using a hardware-based Action Replay or GameShark for the first two, if you have access to an original Playstation with the required port.
Edit: My post on PS2 device pros and cons