I think the next step up from the group of cheap 70-300mm lenses, would be the Canon 70-300mm IS. While the Image Stabilizer does help quite a bit with a lens like this, I believe it also has an upgraded design which makes it a better lens, IS or not.
There is the Canon EF 100-400mm L IS. It's a very big, heavy lens but has very good image quality. It's still 'slow', having a max aperture of F4-5.6 but it's certainly much better than those 70-300mm lenses.
Then there are the 70-200mm lenses. Canon has four models, all with L classification.
70-200mm F4 L
70-200mm F4 L IS
70-200mm F2.8 L
70-200mm F2.8 L IS
All four are fantastic lenses. The F2.8 models work pretty well with Tele Converters, so you could add something like the 1.4X TC to have more range.
Sigma & Tamron also have 70-200mm F2.8 lenses. They are pretty good, but not quite as good as the Canon models. They are about half the price though.
After that, to step up you probably need to look at prime lenses. Canon has 200mm, 300mm, 400mm, 500mm and 600mm lenses that are fantastic but fairly expensive.