In California, anything in the genus Ambystoma is illegal to possess, import, and export. No other genus as far as I know, is banned. I have seen sirens for sale at reputable pet shops who know the laws so I am sure mud puppies are legal.
There are many cool species you can keep in California, just not axolotls. This ban was put into place to prevent larval tiger salamanders coming into the state to be used as bait. Fishermen used to release them into the wild where they posed a threat to the native California Tiger Salamander, Ambystoma californiese.
As for species similar to axolotls, there are legal salamanders that have larval stages that are kind of look like axolotls. If you are looking for a beginner species you could try Fire-belly newts (Hypselotriton orientalis) or Spanish ribbed newts (Pleurodeles waltl).
A search online can help you find the exact laws, if you are still unsure. :happy:
-Advyth