Hi, Panger. Welcome to Rescreatu! We're glad to have you. :D
In terms of getting eggs, it's pretty simple. Just explore the four worlds and pick up any eggs you find. Certain eggs have a higher rarity than others, meaning you will find them less often. Besides common eggs, there are seasonal eggs (which can only be found during certain times of the year) and CS eggs (which must be purchased with cash points in the Cash Shop).
Some other tips: Don't hatch until the egg is at 100%. Regular eggs go old after 7 days and spoil after 14 days. Their hatch rate goes down significantly when they're old or spoiled, but spoiled eggs are more likely to hatch mutant pets.
And as for earning TU, here are two really helpful guides:
If you have any other questions, I'd be happy to help! See you around. :)
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