I'm pretty sure it did autosave in world transition like the other poster mentioned but definitely not the regular autosave intervals that are more common today. I could be wrong about that but I'm not sure why the transition save isn't working for you. Regardless, glad you could get some save functionality working.
Hm, perhaps double check if there's any settings you may have to adjust. Then as the other user mentioned try testing the save-at-dragon-statue. Also maybe try playing /saving a different game just to test if the memory card is working properly too
On what are you playing? OG consoles need a memory card to save to. Sorry if this is known to you but more information is needed to diagnose. (I looove the spyro trilogy, both the original from back in the day, and the remake)