I visited Asbyrgi only once , but it was a superb visit. It is possible to go along both sides of the canyon, east and west, but there is no bridge so it is up to you to choose your own point of view!
Most of the paths are well marked and start from the Visitor Centre. They vary from 1 hour to a whole day with nearly 11 ml of walking.
At Asbyrgi, I myself went on a walk up to Botnstjörn, a little pool with a large variety of ducks, then I enjoyed a lovely walk in the woods, a rare activity in Iceland! In all, the walk lasted about 45 minutes, ideal for getting an idea of the landscape. But of course, the more you advance in the canyon, the greater the number of view points.
Leaving Asbyrgi on foot, it is possible to get to other regions and in this way go on hikes lasting several days with refuge stops along the way.