Translation:You are still young, I am already old but I have never been to Bali.
if i am correct in understanding, 'pernah' translates as 'ever', as in something occurring if even just a single instance or once only. 'belum' negates, but keeps the possibility of an event occurring in the future open. for example:
'apakah kamu sudah menikah?' ('are you married?') 'belum' ('not yet')
'belum pernah' would to me translate something like 'never yet', 'never before' or 'never once'. it indicates that something has not yet occurred, not even once before, but to me, suggests the possibility of a kind of wistfulness or longing without necessarily a strong likelihood, which 'belum' alone does not convey.
the suggested translation in english of 'i have not gone to bali yet' can be sufficiently translated: 'saya belum pergi ke bali'.
'saya belum pernah pergi ke bali' would be better translated: 'i have never before/ never once/ never yet been to bali'.
although it's perfectly possible in english to also say: 'i have never been to bali before/ once/ yet', this lacks the same degree of emphasis.
nonetheless, these are all adequate translations and should be acceptable to convey the sense of 'belum pernah' in english.
furthermore, 'been' should be an acceptable translation of 'pergi' here, in addition to the accepted 'gone'. it is very common in english to say 'i have been to bali', in fact probably more likely than 'i have gone to bali'.