These are main things that you would do.
For most flow:
Paddle upstream:
What you'll do most of the time when there isn't much flow.
Eddy-hop:
Moving between the relatively still/upstream flowing parts of the river - they usually occur around bends at the sides of the river, outcrops rock in the middle of the flow or sides or, at nodes in waves (and the like). You ferry-glide between these until you get where you want to. There's a tome about it in another answer.
Ferry-gliding is a way of crossing flow by using its force to push you across; here's a diagram:
And here's what a river looks like. You can see that the flow is going from to bottom but at certain places it flows back up. You can go up in these and then ferry-glide across to the next.
Below: For Rapids (big and small):
Poling:
This is moving the boat by means of a long pole (about 4 metres long, normally kept in two screw-together parts in the boat). There are two main techniques: using it to push you upstream (or down-stream) by standing up and pushing off the river-bed. This is ideal for getting up short bits that are a bit too shallow to use a paddle with or would be just a bit too hard so you use the pole as it gets more purchase on the river-bed.
like this:
You can also pole in deeper water by pushing it off the water, kind of like a stringy paddle-wheel, but this probably isn't too useful for going upstream.
Lining/Tracking:
These are ways of bringing a boat up or down a rapid by means of ropes, controlled from the bank. One denotes bringing it upstream and the other downstream and y'know, I just can't remember which is which.
Anyway, the general idea is that you get two ropes and attach one to the front and one to the back of the canoe. It's a little like flying a kite. You want to keep the front of the boat pointed out into the middle of the flow so that the water keeps it there and not crashing into the bank beside you, but also move upstream. It's a lot easier with two people but can be done with one. And well, sometimes what you're doing will just amount to dragging.
Back rope: Main pulling rope, pulls the boat upstream, must ensure that it's not let out further than the front.
Front rope: Controls angle of the boat to the flow. The front needs to be further away from the bank you're on than the rear but not so far that the flow exerts more force than you can handle. It needs to be further out so that you can control it with the tension of the rope; if it were nearer you would need to push it back and a rope wouldn't be much use.
Video
Portaging:
I'm afraid that sometimes you might have to get out and walk, lift, drag and curse at the boat until you get it where you want it.
P.S. There's obviously many ways of doing all of these things and there are variations on all of techniques.