Hi and welcome,
Well for starters and since you mentioned is a puppy I know that with kittens airlines won't allow kittens younger than 4 months of age to travel you might want to check that with puppies. Also the weather conditions cannot be too harsh this is either too cold or too hot. (the airline's website will tell you the temperatures limit both for hot and cold to do this)
Most of the information you need you'll find on the airline website, such as: size of the carrier/kennel, health certificate by a licensed veterinarian including his signature and ID practice number indicating that the animal is in good health to travel and with no diseases. Most of the airline requirements will be the same as those of the government in this case of Canada but there will be maybe a couple of extra things like vaccinations up to date with documents to prove it. (I think those are valid for 3 months to travel from the day they were issued)
I think the UK is considered rabies free so you might not need that shot.
Some airlines will allow you to travel with your puppy or cat under the passenger seat (my favorite option) however not all offer this service and I'm not sure at all if any offer it for long trips (like crossing the Atlantic
hence the need to do it during mild weather since your little guy will be unloaded with the baggage :S (I wish we knew better by now)
As for entering Canada with your puppy you should check this out:
You'll probably have to base your flying choice on how pet friendly they are and go for what suits you and the puppy the best.
Keep in mind it's getting colder and also the puppy's age.
If you don't find everything you need on their website (Airline of your choice) I would give them a call as well.
The other option is to hire a company to do this for you. They handle the whole thing and they have deals with planes that do this kind of stuff however it's VERY pricey
someone will accompany the puppy from the beginning to the end of his or her journey and you leave almost everything to them.
Hope this helps a bit~
P.S. When I found about this I talked only to the airline, no cargo stuff as in "hey! I'm traveling with my pet how should I proceed" Regardless of the breed in the end it's a puppy so all dog deals apply to him. It's not like you're bringing an amazonian feathered serpent of 8 arms to Canada