The sentence you wrote is very cute! Let's look at it and make it a bit better.
Original sentence: Just like their mum, baby goats climb trees to eat yummy new leaves.
This sentence is almost perfect! But to make it clearer, we can say, "Just like their mum, baby goats climb trees to eat yummy, new leaves." Adding a little comma helps separate the adjectives yummy and new.
So the corrected sentence is:
Just like their mum, baby goats climb trees to eat yummy, new leaves.
Good job noticing how baby goats do things like their mum!