Learn English – “I’ve no idea…” which is the right preposition

He walked onto the roof.

This doesn't mean the same as "he went and walked on the roof." This means that he walked from another place to the roof. You would be most likely to walk onto the roof from a scaffold or cherry picker or similar. If he was walking while on the roof, you might say:

He was walking on the roof.
He was walking across the roof.

"Onto" expresses movement from one place to somewhere else, where you end up on what you were moving towards.

He climbed onto the tree.

I disagree with Ast Pace's answer on this sentence. While it can mean that you moved to be on top of the tree, I would expect most people to interpret it to mean that you are now holding on to the tree, suspended above the ground but not in the branches. This picture gives a good example.

If you wanted to express that he climbed among the branches of the tree, you could say:

He climbed into the tree.

She walked into my room.

You are correct on this sentence. It means that she entered your room by walking. However, you wouldn't say "entered in" your room – the room is what is being entered, not something inside your room.

Where're you going for your holiday(s)?

Where're you going on vacation/holiday?

You normally use the preposition "on", but the use of the "for" isn't ungrammatical.

As for the sentence "Tell me a good place to go on holiday", there's nothing with it grammatically.