The following information refers to UK passport holders.
It's pleasingly easy, in most cases, to arrange a visa for North America, especially if you're only going as a visitor. If you're travelling to the USA you'll need to register online at least 3 days before departure, and a tourist visa is valid for up to 90 days. For Canada your visa is simply stamped upon arrival and lasts up to 6 months.
If you're travelling to Mexico you need a tourist card, which you can get on arrival by completing an immigration form. You must hold onto this as it's required when you leave the country.
The majority of Caribbean countries do not require you to get a visa, but this is definitely something to check ahead of time, as requirements do vary from island to island.
Things get a little trickier if you're looking to take a working holiday in the USA. You must be a full-time student or have graduated within the last 12 months to be eligible for a working visa, and this can take up to 2 months to process. If you're going to be working at a camp you will only be allowed to work at your stated camp, though these visas also allow for up to 30 days of travel around the country.
Canada, perhaps not surprisingly, is a little more relaxed. You can get a working visa (IEC - International Experience Canada) that entitles you to stay for up to 12 months, as long as you have a written job offer or contract of employment from a Canadian employer.