Coming from the UK you will be surprised at how vast and varied the country of Canada is. For a general overview of weather and travel you could check out:
Canada Weather and Event Calendar - Canada Weather and Events by Month
When I traveled to the UK I remember how bleak and gray the country seemed at the end of winter in March with bare deciduous shrubs and trees , and how spectacular the change was as winter ended and spring began. By late April and May it is like a different place as the trees leaf out and spring blossoms forth. Canada is no different except that spring comes a bit later and faster.
Of course these are sweeping broad generalizations which is all I can offer for such a big question.
Personally, if I were planning a trip to North America and was planning to land in Toronto I would land in May or June and head for the maritime provinces of Nova Scotia, New Brunswick, Prince Edward Island, and Newfoundland. Lots of backroads and byways. I really liked the winding Cabot trail road on Northern Nova Scotia and the friendly Canadians I met everywhere I went. That could take a few weeks or months in itself.
Instead of heading across the vast prairies of Canada I would take the ferry to Maine and explore the New England states and the northern tier states in the summer when the weather is warm spending more time in Colorado rockies, Utah, northwest Wyoming with the wonderful Tetons, Bighorn range, before heading over Beartooth pass to explore Montana, Idaho river of no return, Hells Canyon......
Okay, I am typing this, reliving all the beautiful places I have been on a motorcycle and I realize the folly of trying to do this in six months.....I was thinking head north to Banff from Montana and hit the icefield parkway to Jasper and all the wonderful roads in the Kootenays or heck why not cut across from Jasper to the Cassiar Highway and head to Alaska.
But you get the idea. Northern area in the summer, southern states and desert southwest in the colder months.
Basically land in Toronto in late spring, hit the Maritime provinces, drop down to New England, definitely hit western North Carolina, the Cherahola skyway, Deals gap and all the great roads in the smoky mountains, jet west to the rockies and don't miss the Pacific northwest, California and the Grand Canyon...
Dang it, now I want to go with you.