The best place to look is definitely the skull cavern, where those pepper rex dinosaurs show up on the prehistoric floors. They've got a pretty decent chance of dropping eggs, and you might even find some just lying around while you're exploring. Just make sure you're prepared with good gear and bombs since those dinos can be pretty tough.

If the cavern feels too challenging right now, you could try your luck with fishing instead. Once you've hit level 2 in fishing, you can find eggs in treasure chests, and using magnet bait or treasure hunter tackle will boost your chances. Maybe interesting: eating some luck-boosting food like shrimp cocktail can help too.

Those wiggly spots in the mountain area and quarry can sometimes give you eggs, though the chances are pretty low. These wiggly spots tend to show up more often when it's raining, so that's worth keeping in mind. There's also this neat little trick with the crane game at the movie theater, it's not the fastest way but it's there if you want to try something different.

Whatever method you choose, when you do get your first egg, definitely pop it in an incubator instead of giving it to the museum. That way, you'll have a steady supply of eggs from your own dinosaur which is way more useful in the long run.