If you're used to playing live service games, you should already know that you don't need to worry about saving your progress. The same is true with Wuthering Waves, as your progress is saved real time; this means you can abruptly close the application and return to it to find yourself in the same exact spot as you left it.
Wuthering Waves also allows cross-platform play, which means your progress is retained regardless if you're playing on mobile or PC. As for my personal experience, I play primarily on the PC - which I personally say offers the best experience. I also have the game installed on my gaming laptop but then again, having the ability to just log in using my mobile phone and do a quick run like completing mundane tasks that doesn't require too much combat is a pretty flexible and convenient option to me.
So how do you enable this feature? Basically, you just need to use the same login details to access your account. The seamless way to do this is by using the same email address. You should also take note that your progress is kept within the server. For example, you started a game in the Asian server. On your other device, after successfully logging in, make sure to select Asian server as well. If you mistakenly selected a different server, the game will start a new game but you can just log off, enter your credentials again, then select the right server to continue where you left off.
Right now, the game is only available to two major platforms: PC (via Windows, Epic Store, and Google Play Games - which is basically an official PC platform that allows you to play mobile games on your PC) and mobile (via Appstore and Google Play). The developers stated that they do intend to release the games in the consoles and Mac. However, there might be some caveats and technicalities regarding this. As far as I know with the other similar titles that did this (like Genshin Impact and Honkai Star Rail for the PS4/5), you'll need to have a corresponding account for those platforms (like PSN) then link the email that you used for registering. Right now, until an official statement has been released, this is just my speculation.