Do I have to save my private keys? Is it a good idea to take my private key from my wallet and save them? Because I read it is not save to make your wallet shows your private keys. (It guess I do not understand that well).

Edit: Are they different than seeds? the words-phrase that wallet gives?

And how can I use the private keys to restore my coins in case for example, my wallet company stopped working?
Yes indeed you should back up your wallet, printing out a paper copy of your private keys is a good way to do this because paper can only be accessed physically rather than being hacked via the internet. Obviously it's not safe to store them unencrypted on the cloud or something. The security warnings are there just to make sure you don't do anything stupid with them, to make you double check before potentially giving others access to them.

Bitcoin Core for example allows you to import private keys to restore a backup. If you print them out as QR codes, many wallets allow you to sweep them