If you have a PS4 and you want to play your PS4 games on your new PS5, that’s pretty painless. Using either cloud saves or the system transfer, you just move everything over. Running the PS4 versions of your games on the new hardware, all your progress should be there.
As far as I understand it, PS5 versions of games are seen as completely different games than their PS4 counterparts by the system (I think this is where the Xbox handles it fundamentally differently but I don’t know for sure), and a PS5 game won’t see PS4 saves at all. My PS5 copy of Spider-Man could no more access my PS4 Spider-Man save game than it could access my PS5 Control save file. Games don’t see other games local saves (or saves in Sony’s PS+ cloud save storage) period.
But there are games that allow your progress from PS4 to carry forward on the PS5 through one of two workarounds: either by always saving your progress to their own servers, or by including in (or patching into) the PS4 version a one-time export of the save file to their servers where it can be imported by the PS5.
Games like Fortnite and Destiny fall into the first category, where your progress always lived in the server, and so these are the games that maintain your progress “automatically”. These tend to be more multiplayer focused games.
Games like Spider-Man fall into the second category, where the feature to export a save game was expressly added for this generational jump.
Doom Eternal is a weird example of these different approaches, because some element of your multiplayer progress did cross over because it was always saved on the server side, but no export was offered for the single-player save progress.
I would not count on many more games with PS4 to PS5 transfers in the second category; this feature seems to be quickly disappearing, not becoming more popular.
Googling didn’t turn up a definitive or recent list of games that let you move your progress from PS4 to PS5, the best I could find is a partial list from December.
The bottom line is you shouldn’t assume this functionality. Look into it for specific games you care about.