Say I'm a sailor on a ship bound for Kingston, Jamaica from Nassau, Bahamas. How much time would this journey take under ideal conditions? Weeks? Months?
Your distance is about 670 Nautical Miles. The first leg is close to the wind (given the prevailing trade winds) until you round the East end of Cuba. Your second leg is downwind (but is a shorter leg).
If your ship averages 4 knots across both legs, your journey time will be about 7 days.
Rather specific question - I smell something related to Golden Age of Piracy here. Would love to know where this question got its inspiration from.
Anyway, it should not be a long voyage since you could take the old trade winds pretty much to it. I would say this voyage is something to be counted more in days or a low number of weeks. Let's say is a good ship that is averaging about 6 knots across the greater voyage (since you said ideal conditions, I see 6 knots as real good, especially considering some of the time required sailing somewhat against the wind to get up into and through the Windward Passage would put you below 6 knots). Plotting the likely voyage from Kingston through the windward passage between Hispaniola and Cuba and up into the Bahamas to Nassau would be about 700 miles. 6.906 miles per hour (which is what 6 knots translates to) into 700 miles would equal 101.36 hours of constant sailing, or little over 4 straight days. 5 to 7 days sounds pretty reasonable from what I've seen.
But, you have to remember, it's the Age of Sail and we are dealing with sailing ships with no modern communication systems. That ship could probably go two weeks and probably no one would be surprised since there are all kinds of ways a ship might be delayed. Weak winds, storms, contrary winds, accidents, dealing with other vessels (be it pirates, the guarda de las costas, Navy search/contact for who knows what, a civilian ship flagging you down for assistance (or just news), and a host of other reasons. There is a article I have about how quickly news of Queen Anne's ascension to the thrown of England traveled that is good in demonstrating how quickly ships could move about the Atlantic (and the Caribbean) - but I don't have access to it this minute so I'll do a edit later to add it to this post.