Hello. Is it correct to write in to or into, and on to or onto?which is more common?

Both into and onto should be spelled as a single word most of the time.  They are prepositions that tell 'where'.

He climbed onto the bed.

She peered into the bottle.

However, there are times when "in to" and "on to" are correct.

We looked in to see what the children were doing.

In this sentence, the word 'to' is part of the infinitive form of the verb to see.

He went on to explain the meaning of the book.

Again, the word 'to' is used as part of the infinitive.