Animal eggs are just regular items, they are not traded in like tokens, so their only use is just as an item to collect or to decorate your pets. They are considered a bit more valuable than most items since they can only be gained through the banner during the Easter event, and they tend to sell for more during the event and especially while the eggs are still new and not many people have them. There is plenty of opportunity to get them from the banner throughout the event though and they typically decrease in value a good bit by the end of the event.
Special release pets are primarily what we also call UR pets, these can only be adopted during the Advent, on Dec 18th. There is a small chance to adopt any previous or new UR pets from the re-release on that day. They can of course be traded for from other users throughout the year, but they are some of the rarest pets on the site (they all have OMGsr rarity) so they are quite expensive. You can find out more about UR pets
here.
Events are a great time for newer users to get rares since it doesn't require having any pets beforehand! During token events (Easter, Summer event, and Halloween) many of the pets you trade tokens in for will become rare. The more popular ones usually end up with a lower rarity as it is dependent on how many people traded in for them. The free adoptions on the event page do not usually become rares, just the token pets (they are still good to adopt, just don't expect them to be rare). If you would like to get rares I'd just suggest focusing on getting as many tokens as you can and trading those in on the Event page. The pet rarities for the Easter event usually isn't released till after the event is over so there is a bit of a wait to find out which pets will be rares or not.
Monthly litters can sometimes have rare outcomes as well, but this is not especially common right now, and there is a chance you wouldn't get the pet that ends up being rare. Events are definitely the best guaranteed way to get rares. You may also want to check out the
Free Adoptions board, sometimes adoption centers will have giveaways specifically for new users and give out higher rarities during those. Uncommons and lower rarities do slowly move up in rarity over time, but this is an extremely slow process (often taking years) so its not especially helpful for new users, I wouldn't bother holding onto pets waiting for them to go up in rarity.
