Apart from making your way round the Carribean or Polynesia by offering a hand onboard yachts which seems to be common and easy, the most common route is the Atlantic crossing from Europe to the Americas (mostly to the Carribean or Brazil).
East-West from Europe
When to go: Boats go with the trade winds that start to move from East to West across the Atlantic in autumn. So the season is from September to January-February. Top season is October, November. In the beginning of November each year there is a regatta -the Atlantic Race Crossing- leaving from Gibraltar. There will be more boats than at any other time in the marinas and it can be considered securer than leaving with a boat that is going unassisted. There will be more competition on finding a lift though, too.
Be aware that for the past three to five years the winds have started to go haywire a little bit, acting less predictably with more storms happening. This is most probably because of global climate change. The way most sailors comment on this is "the winds are going through a transitional phase to find a new rhythm".
East-West from Africa
If you don’t want to pay the ferry in Algeciras it is possible to sail to Africa, although except you are extraordinairily lucky, you’ll have to go via the Cap Verde islands which are a stopover for many trans-atlantic sailors. They are between 10 and 14 days from Gibraltar. From there you’ll have to catch a new boat to get to Senegal which is three days away. It is an experience in and of itself although maybe not the perfect swap for a one-and-a-half hour long ferry ride that’ll cost you 25 euros You also miss out on Morocco and the crossing of the Sahara dessert, which are highlights of any visit to Africa.