Apart from making your way round the Carribean or Polynesia by offering a hand onboard 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).
Atlantic Crossing (Europe to the Americas)
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".
Best ports to hitch from to cross the Atlantic:
Starting from France: South: Antibes North: Brest
Southern Spain: If you are English speaking: obviously Gibraltar. Also Malaga.
Canaries: The biggest marina is on Gran Canaria, in Las Palmas.
(Morocco to the Canaries: Essaouira, you can try Agadir also.)
Senegal: Dakar and Casamance (see "Africa to America" further down).
Experience: Experience in sailing is not necessary -although a huge plus in getting a quicker lift-, but participating in duties and life on board of course is obligatory!
Some money: These days most yachts will ask you to chip in for your food. Most boats ask either 10 or 15 euros on food per day. The crossing takes between 15 to 25, depending on boat and winds, so you'll need around 150 to 300 euros for the crossing alone -count on at least a week in the marina till you find your lift, too. It is possible to get an entirely free lift food included, mostly on bigger yachts where you'l be needed to scrub the deck and polish the silver railing. During the ARC you can even expect boats ask as much as 50 euros per day since inscription fees are pretty high.
Europe to 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 make up highlights of any Africa traveller.
Africa to the Americas
From Senegal: Dakar or Casamance. With a very likely stopover of the Cap Verde islands.
If you want to take a flight to the Cap Verde islands: the biggest marina is in Mindelo on Sao Vicente, the second biggest one is on Sal where the international airport is also.
Since sub-Saharan Africa is out of the influence of the trade winds you can hitch from Senegal almost all year round, although the main bulk of boats will be leaving when it is top season in Europe, that is October to December.
There are three sailing clubs in Dakar. “La Voile d’Or” with shallow water which therefore attracts catamarans only, the “CVD” (Club des Voiliers Dakar) where the majority of boats can be found and a third one also in the vicinity of the two ones mentioned. They are all situated not far from another to the South of the “Cap Vert” peninsula and the town centre. If you try to hitch from there it is strongly recommended you speak some French.
There are stories of people who always know someone else who managed to go for free, but the only reliable stories really date back to the seventies. Cargoship travelling is commercialised now virtually everywhere. For something like 50 pounds per day you can rent a cabin on them.