Correct way... I rode home on my new bicycle.

In my opinion, the verb "cycle" is not appropriate in your sentence and seems redundant with the existence of the noun "bicycle". maybe you can replace "cycle" with "pedal" "I pedaled on my new bicycle back home." / "I pedaled home on my new bicycle." (nonetheless, I also agree with Echo) :)

You're looking for "on" - I cycled home on my new bicycle. Echo has made an interesting point, I think the main objection is using similar words too often (cycle/bicycle) and that the verb 'to cycle' implies you are already riding a bicycle. Her suggestion of 'rode' is good, or you could also say, "I cycled home on my new bike" because it is significant that the bicycle is new.

I dont think it's true to use that bicycle word with the verb ' cycle'. It just includes it. It'd be enough to say ' I cycled home' or if you wanna emphasize 'the new bicycle' ,"I rode home on my new bicycle " seems OK to me.

Hi... Prepositions are very important in a sentence. Check this to understand better if you want : http://www.englishpage.com/prepositions/prepositions.html