"Our youth like to travel."
Translation:Наша молодь любить подорожувати.
I believe the usage of singular/plural verb forms after nouns denoting groups of people is one of the differences between British and American English. See http://blog.oxforddictionaries.com/2011/09/agreement-over-collective-nouns/ for more info.
Both should be accepted.
Молодь is a feminine word. When a word ends in a soft consonant, it can be either feminine or masculine.
The fact that the word denotes a group is irrelevant in Ukrainian: if it’s a singular form of the noun, then you need to use a singular form of the verb, too.