Translation:The little green frog climbs up onto the shelf and jumps down onto the armchair.
The English used in these lessons is sometimes awkward because the writers are trying to show a sense of motion that English doesn't really have. As a native English speaker, I wouldn't talk this way, but I understand what they're trying to do. Hungarian grammar indicates things English doesn't.
I would like more accuracy here. I am not a native English speaker and many times I have more difficulties in trying to translate sentences from Hungarian to English than vice versa. I have talked with a native English speaker who said that in many cases "onto" and "into" are optional. Many times one can use only "to" without a difference in meaning.
I think what the native speaker meant is that "onto" and "into" are sometimes interchangeable, not that they're optional. "Onto" and "into", despite meaning different things, would strangely both be correct to use here, but "to" definitely isn't, unless the armchair is sitting on top of the shelf.
I disagree, "jumps down to the chair" sounds totally reasonable to me (and I am a native English speaker). It's less specific (it could mean merely that the frog jumps near the chair, rather than specifically on it, but probably means on in this context anyway). And I can't see how the distinction between "onto" and "to" has anything to do with whether the chair is sitting on top of the shelf?