The thing with cargo ships is something that cannot be fixed with a "hey, go to this website and..." or "hey, this is a company that..." unless you REALLY want to avoid flying and have some spare thousands. I have been heavily checking prices for the service that cargo ships offer to take you places, and they are VERY expensive. Im talking about 2000 dollars as a base or even more for most trips, including the one you are hoping to get. But dont get disappointed on the subject already, because the trick is to find your way trough for free or even get paid for it. I heard stories of people hanging out in bars near ports and getting friendly with freight people that hired them for seriously long trips where not only they got taken places but also got paid for it. Also there's the chance of doing some small job and convince them that you can sleep rough and then go for free while justifying your presence on the ship, and a long list of very imaginative etc.
About sailing boats, I have lots of experience with the seeking of transatlantic sail boats to hitch with. And its pretty much the same as above. Cant say "go to this website and..." or "ask this people and....", but its possible to say that. There are a bunch of websites that connect boat owners with potential crew members, there was one in particular that had a huge user base but the webmaster got greedy and started asking for money to have an account and later on even for each individual connection (meaning that everytime you want to ask a boat owner, you have to pay a single fee) which totally sucks. Still there are some free websites that do the same, but they are not so popular. Anyhow, the best way to seek a sailboat ride, is to identify the ports where such rides originate, set camp/base nearby and everyday hang around them boats, talk to people, tell everybody you intentions of crossing the Atlantic. Make sure you have something to offer. If you know some sailing you will most definitively get a ride, so if you got some friends who have a sailing boat, ask them to take you on some trip with them to learn the basics. No need to get crazy on it, good enough sailing knowledge can be learned in a matter of days if you are serious about it, and that is handy enough for any boat captain to want you in (as you will not be expected to be a captain but a mere substitute when he/she wants to rest or the weather is easy).
Unfortunately, I have no idea where America (continent) to Europe Sailing boats originate and which is the season, but im pretty sure the info is out there if you look it up. I suspect stuff happens around the Caribbean. I can definitively help you come back but I promise lots of competition, as sailing boat hitchin' is getting more and more popular for euro travelers and the spots are getting crowded each year, creating some sort of market opportunity which makes boat owners get greedy and start asking money for rides (since the fucking hipsters have it and wont mind paying lots to say the "hitched a boat").
There is another alternative you can check, and that is passenger boats, which have become more and more luxurious as air travel became so popular. There are some boats NY / England (back and forth) with the queen Elizabeth boat and some others. Of course you should expect it to be very expensive, but look for a job there or some kind of connection, similar to a cargo ship and get on them. Those ships used to be very difficult to get jobs on, because they used to pay very good money, but I have heard that now they are pushing very low money on their workers making workers rotation faster and faster, which gives the traveler a chance to get a one ride job with no hassle. So look it up as well.
The last one, if none of this works or if you just want to get there, look at a globe, not a flat map. Realize that Toronto / NYC are VERY close to Reykjavik / Edinburgh / Oslo (Iceland / Scotland and Norway's capital), so flights to those destination's can be insanely cheap, on the 100/200 USD range if you are good at searching flights while patient enough.
Hope that helps