Dmitry do you have a good dictionary at home? There is just addition to Alan's examples. We got onto the train. My friend got onto a horse (сел на коня). His sister jumped onto a tram (вскочила в трамвай). Put me onto your ideas. - Расскажите мне о ваших идеях.

But you are wrong, She climbed onto/on to the roof, and Let's go on to the next chapter (you cannot use onto here).

If it helps, 'onto' just means 'on to'.

I threw my clothes onto the floor. The dog jumped onto my bed. The satellite fell onto the Siberian plane.