I am very confused about which one I should use in which situation.
For example:
Let’s look through your closet and I’ll help you pick something out.
can we replace out in exchange for up here.
Another example:
A child go to a patisserie with his mom and there are some different kinds of cakes available there. But the child can only buy one of them. So what would his mom say?
Just pick up the one you want most.
Just pick out the one you want most.
Just pick the one one you want most.
By itself, pick (in this sense) means to select:
Look at your choices carefully and then pick one.
Pick out means to select from a large group (at least, it does in my American experience).
She looked at a lot of dresses and picked out the one that she liked the best.
It can sometimes mean detect something from a collection (i.e., detecting something based on observation, rather than choosing something that you like):
Can you pick out which shape is bigger than the others? (In this case, pick alone sounds strange, because you're choosing a correct answer, rather than making a choice based on preference.)
The crowd was very loud, but I could still pick out the sound of my friend's voice twenty feet away.
I police asked me to pick out the thief from a lineup of suspects.
Pick up is a physical action meaning to lift or collect. It is not a mental action. It has nothing to do with selection.
I'm not very strong. Can you pick up this heavy box for me?
You should pick up the trash that you dropped on the ground.
I will pick up some beans at the store.
Can you pick up John after school?
This meaning of collect can also apply to ideas or knowledge to mean learn:
I picked up a little bit of French while I was in Paris.
I picked up how to use a screwdriver from my parents, who are carpenters.
After his third suggestion that I stay home and rest, I picked up on his meaning: he didn't want me at the party.
In your case, pick or pick out would be fine. Pick up would not be correct, unless you are telling the child to physically lift the cake with their hands.
Pick means select one from available choices and take it for yourself as a decision.
Pick out means pick the odd one out from available choices but don't or can't take it.
Pick up means pick something which is lying somewhere as in location or place. There may or may not be other choices. It also means changing the state of something as in learning and improving, like he picked up football quite fast.
She would say "Just pick one you want most" in this particular context.
I hope I made it clear.
