Is this Playstation 3 backwards compatible for the PlayStation 2 and 1?

I would like to play the old classics like God of War and the Souls games, but I need a console. Am currently looking for a console that can play the most games. Can someone tell me whether this Playstation 3 is backwards compatible or maybe give me an alternative.

LG let

The PS3 can generally play all PS1 games, but you are very unlucky with PS2. There are very, very few PS3 models that can play PS2 games, and then not all of them.
https://www.quora.com/What-was-the-best-model-of-PS3-for-backwards-compatibility
As you can see in the picture, the very first PS3 models are compatible (with 20GB storage and 60GB) and those with 80GB storage space. But whether you can still get one of the models today is questionable. I think it is more worthwhile to hunt down a PS2 right away.
Otherwise there's still the possibility to emulate PS2 games on the PC, but this is a legal gray area and downloading the ROMs / games themselves is illegal.

Thank you, do you happen to know whether the Ps2 is sufficient to play the old God of wars and souls games or do I still need the ps3

If it's just the two franchises in particular, a PS3 is sufficient, since God of War has Remaster / Collection for PS3 and Dark Souls only started in the PS3 era.

The PS3 can generally play all PS1 games

Unfortunately, it is not 100% true. Try to start the Pal version of Tombraider 1 (the one from 1996), more than the start menu will not work

Thank you very much. Because you mentioned emulating once. Does it work properly, or does the game buggen?

Partially. It often depends on which settings you make in the emulator and depending on the game. The whole thing is often a thing in itself, but there are now some instructions (more in English) where it works quite well and the communities are quite helpful.
E.g. Epforums.org still exists (even if half asleep) and otherwise there are FB groups and chats for emulators.