The Tourist Office offers excellent guided hikes throughout the summer - free with the Joker Card. The hotel will have the weekly programme of hikes. Even though you have downloaded maps be sure to pick up a free hiking map at the lift stations - vital if your phone runs out of power!
The big hike from Saalbach to Zell am See is a hard-core hike that takes several hours. You need to catch the early morning "hikers special" cable car at the Schattberg Xpress which runs a few hours before the lifts officially open, otherwise you might be pressed for time.
Easy, interesting hikes:
Take the free bus to the end of the valley at Lengau, then walk past the Lengauhof hotel, through the car park and the keep the river on your left hand side as you hike up to the Lindlingalm and the high ropes park. Continue through the farmyard and water park and then walk around the head of the valley clockwise by crossing the river just after the water park, then walk up past the Saalalm and the Ossmannalm huts - good food and cold beer at both - before returning back to the Lindlingalm.
Ride up the Reiterkogel lift in Hinterglemm, then follow the easy trail past the reservoir to the Rosswaldhutte hut. From there, either continue on to the Hochalm area (from where you need to walk down) or retrace your steps and follow the route above the reservoir back to the lift.
Ride up the Kohlmais cable car in Saalbach. At the top, walk around and under the lift station towards the chairlift (lots of marmots just there). Find the Wurzel Trail behind the chairlift station and follow the hiking path along the ridge towards the Schonleiten lift. For a longer walk, just before the section up to the Wildenkarhutte, take the path to the left to the Schonleitenhutte. From there, there is a path that leads to the top of Asitz in Leogang. From Leogang you can take the bus back (the bus stop is on the main road at the Asitz lift) via Saalfelden (change for the #260 bus towards Zell) and the Maishofen crossroads (change for the #680 to Hinterglemm). The buses are timed to meet each other.
Take the Schattberg lift up and turn right. Walk down to where the paths split. Take the left-hand path (over a stile) and walk around the path until you're below the Westgipfelhutte. Either continue uphill to the hut and yellow Westgipfel cable car, from where you can ride down into Hinterglemm, or walk back to Saalbach using the wide path next to the hut. Or - below the Westgipfelhutte there is the Hacklberg trail (there's one for mountain bikes and one for hikers). Follow this to the Hacklbergalm, then back to the mid-station of the Westgipfelbahn. (You can of course do these walks in reverse, starting by riding up the Westgipfelbahn. Do not try to walk down the Schattberg).