Does 'on' preposition go with the verb 'take ? or should it be
'I would like to take up new challenges'
which one of the two statements is correct ?
Both sentences are correct, and both can be heard in everyday American speech. Some of us speakers are anal and might insert a hyphen as follows: take-on & take-up. It's not necessary to insert a hyphen, but it emphasizes that the two words go together and are used together. It's like transforming the verb "take" into a new verb "take-on"/"take-up." However, they really are two, separate words, and it wouldn't look right if they were combined outright such as "takeon" or "takeup." Again, it's not necessary to insert a hyphen; it just makes the sentence look polished and cleaned up (or "cleaned-up"...lol).
The difference between "take on" and "take up" is thus: imagine "taking something on" like a football player rushing towards an opposing football player...a direct confrontation/direct conflict/direct challenge...and imagine "taking something up" like carrying a heavy box or heavy package...or the Greek god, Atlas, "taking-up" (or "taking up") the weight of the world...a challenge. Both roughly mean the same thing.