I always played games growing up but didn't play Kirby until much later in life. Kirby is easily my favorite platformer, even ahead of Mario and Sonic. I wish there were more like it.
Super Mario Land is a good place to start. The music is great, and it is very short and pretty easy.
Link's Awakening is my favorite Zelda game.
The original Pokemon games (Red/Blue, Gold/Silver) that others are recommending are pretty rough these days, but it's fun to see where it all started if you have the patience for them.
If you want something a little less well-known, Bubble Ghost is a fun little puzzler.
Regardless of what you try, remember that this is the era when game manuals were required reading since they rarely were able to fit tutorials & instructions in-game. It's good to scan through the manual first before you play something or you might be missing some critical information.
There are definitely limitations, but later games have a lot more depth than these.
Link's Awakening feels like it shouldn't be possible in a single megabyte of mask ROM, for example. The fact that it was makes it almost more impressive by today's standards.
I feel like I don't really encounter stories of heroics in making games work on their hardware anymore. Even at the end of PS1 with Final Fantasy their hand drawn backgrounds were too big to fit into VRAM iirc
Or in the PS2 era Jak and Daxter using the principle of hot reloading from lisp to figure out open worlds without loading screens
The fact that a single dev used to be able to complete an entire game really makes me wish I had come of age back in the day when that was possible. I feel like you can only approach something similar with pico-8 or another fantasy console these days.
Ah, right, I fortunately slept on it long enough to get the DX version right away! The photographer side plot was amazing, especially with a Game Boy Printer :)