Actually, you cannot currently explore Uldavi!
However, there is a quest where you can get the eggs! Under games->quests at the bottom there's kirs quest. You hand in pets he asks for and get points for them! Higher colored pets give more points. Once you turn in a creatu it disappears forever though so don't turn in a favorite if you wanna keep it..
It can be very discouraging and difficult for new users, so if you do start it, be patient, because it can take a while if you don't have a lot of tu or creatu!