Strictly using a map and compass is not what the beginner should be doing! Use a GPS! However, if you have a GPS with you and you've set it up correctly, it's a safe way to learn.
It's essentially the same as using a GPS. You still need a good-scale map, you still set waypoints. You need a good quality compass! I use a Silva Trekker Compassmodel 420 to be specific. It has a inch and millimeter ruler on either side, a rotatable dial, and a mirror that makes sighting easier. It's small, durable, and gets the job done. There are a lot of cheap compasses out there, be wary of them. I've had them fail on me and fall apart.
To navigate with a map using a compass, you're going to have to find the difference between magnetic north and your map's grid north. Compasses don't point North, they point to the magnetic pole which is not fixed. There's a bit of a difference between the two which varies depending on where and when you are.
Most maps will have grid lines along North, South, East, and West, but they are not the same that your compass will indicate. To make it easier, a lot of maps will have a set of arrows that show you the difference; you can place your compass right on it and adjust it visually. If you don't have that, you can find your magnetic delcination online. Keep in mind that "grid north" is also not always the same as "true north". It's a bit wonky, but your map will at least tell you the difference between those two.
The pictures below illustrate aligning your compass with grid north. See the "magnetic north" arrow? Rotate your map until that's lined up with the solid red compass arrow, like in the second picture below. Then, turn the dial of the compass until the bottom, transparent arrow is pointing towards grid north. After this is done, your compass arrow will bit inside of that bottom arrow when you are physically facing grid north.
Aligning Your Compass With Grid North
After you have your compass calibrated to the map's grid north, you can orient your map so that it's lined up with the real world. This will allow you to eventually sight your heading. Align two sides of your compass perfectly with two grid lines. I've drawn the grid lines I'm lining up with in red. When the map is lined up with the real world, the red arrow of the compass will fall within the red bottom arrow, like you can see below.
With the map oriented with the real world, you can set your heading on the map, like in the picture below. Find out where you are and where you want to go. When you're canoeing, this would be a waypoint that is distant but still not blocked by trees or anything like a point of land jutting out. In this case I'm sitting at my current campsite in the Kawartha Highlands and pointing to the one across the lake for practise.
Place the side of the compass so that it makes a line that touches both points (where I am, where I'm going). Make sure you don't move your map while you do this! When you have the two points lined up, rotate the dial of the compass so the dial arrow falls in with the compass arrow.
With your heading set, you can point the compass to find it in real life! The compass arrow will fall within the dial arrow when you are pointing straight at your target. Use the mirror so you can sight it directly without taking your eyes off the target.