When you are out canoeing, swimming is an integral part of paddling. Everyone paddles until the next tip-over-and-swim. This means that everybody gains from good swimming skills. Self-rescue might sometimes be  your only option, so it's not always an option to rely on your paddling partners to get you out.

Roughly there are 2 distinct variants of swimming in a river: with gear and without gear. And of course there are variants where you only have part of your gear in hand, or maybe another swimmer or ... .

Try to ignore the reflex to stand up as soon as possible. A safe rule of thumb is that it's not safe to stand up in current where the water level is at or above your knees.

Swimming without gear

Fundamentally there are 2 positions to swim and keep you relatively protected from foot entrapments and other risks. Several names are used for the options, but both have advantages and disadvantages.

Defensive swimming

In this position you lay on your back, feet high at the water surface. Try to be as flat as possible on the water, so don't raise your head too much. If you do that your butt drops in the water and you could get hurt like that.
Very important here is that your feet are downstream of your body. This way you avoid hitting rocks or other immutable objects in the river head first. You can even buffer the impact or change direction by pushing off with your feet against the object.

Swimming in this position allows you to keep a very good overview of what's coming downstream. You can line up your body in the current and make a ferry-like manoevre just the same way as you do when you're in your canoe, without losing the overview.

An Example:

Agressive swimming

This way of swimming is often called agressive swimming. It is anyhow more active, since you turn on your chest, and swim in an active crawl towards a given point. Feet and body are high at the water surface and thus relative safe for foot entrapments.

Since you're facing down, this way of swimming is not optimal to keep an overview of what's going on downstream of  you. This way of swimming is also pretty intensive, so probably not recommendable for doing a long time.

An Example:

Mostly in case of a swim, you turn into the defensive position, look around, choose an eddy or safepoint and start lining up in that defensive position. As soon as you are coming within reach of the safepoint, turn to the agressive swimming to reach the safepoint.

Swimming with gear

I've mentioned there are several variants here. But let's assume you have all you gear with you: canoe and paddle. I won't go into the discussion if and how long you attach ropes to your canoe. I tend to  have ropes on my canoe for about half a boat length, both on bow and stern. I use these ropes for swimming. But most paddlers have a problem handling their paddle when swimming.

That's easily solved, paddle with it. That's why you have it.

I take the rope in my grip hand, and hold the paddle in a normal paddling position. Line up your body for a ferry like maneuvre, and paddle backwards with your paddle. You can add extra power by working your feet.

The next image shows the grip position:

And here a snapshot of a swim: