These are just a few suggestions, the owner is happy to provide more ideas!
Drive, take a bus or train to the Gotschna cable car (max 10 mins). Take a cable car to the top station. Walk the "Panoramaweg" from the top of Gotschna to the middle station of the Parsenn funicular over Davos. About 2 ˝ hours, dropping about 700 metres. Take the funicular to the bottom or walk ( a further 45mins) train back from Davos to Klosters Platz or Dorf. It's a spectacular 15 minute train ride back sweeping in tight curves on a sharp descent, seeing Klosters first one side of the train and then the other. A gentle first day. The walk can be extended by a further two or four hours by talking either the Hoheweg from the top of the Gotchna cable car station to the top station of the Parsenn funicular, or climbing up to the top of the Weissfluhgipfel (a further 300m higher) and descending from there. Hard effort, but rewarded with spectacular views.
Walk from the Chalet through the forest that begins at the rear (it is yet another protected nature reserve) up to the top of the Madrisabahn (or save energy, walk the 10 minutes to the bottom of the Madrisabahn and take the bubble car up. From there two very different walks are possible. To the left, walk on the "Energy Walk", a signposted walk across water divination spots to the ancient granite outcrop of Madrisa itself. The granite is much older than, and a much lighter colour, than the surrounding area. From there drop down towards Saas or Kublis. A long, gentle descent of 1000 metres (2 ˝ or 3 ˝ hours). The walk takes in constant views of the Prattigau valley, descending through meadows and small Alpine hamlets used by farmers in the summer for high grazing, before descending to the wooden villages at the bottom. Alternatively, turn right and walk along the top of the ridge to Austria. Lunch at the top of the Schlappinjoch pass, just inside Austria. Descend and walk beside the long, tumbling stream all the way back to Klosters Dorf. Be sure to stop at Erika's Gasthaus on the way and look around inside whilst enjoying drinks and the speciality nut cake.
Drive, or take the train and bus to the Rinerhorn. Either walk up (2hrs) or take the bubble car. Completely different scenery and views again. Views down the Davoser valley to Filisur and the famous Albula valley. . Either walk down (2hrs), but, more fun, descend gently through the Sertig valley. It's the only area that is full of heather and gorse - more like chalk downland than high mountains. The walk to the village of Sertig itself takes about 3hrs. Sertig is another tiny wooden Alpine village, so pretty that a modelling firm has produced the entire village in kit form! There is another lovely café there, a great place to watch the cows being herded down the middle of the main street to the milking parlour, cowbells clanking away. Either walk back to Davos (2hrs) or take the postbus. This is a treat in itself as it careers round sharp bends on a road wide enough only for the bus. The risk of traffic coming the other way is dealt with by a special three tone horn played at regular intervals. The law says that when the postbus horn blows, everything else must get out of its way!
Bus or car to Monbiel. Walk beside the rushing Landquart River, past falls and islands, to Alp Garfiun for lunch. Enjoy the sun terrace, then cross to the other side of the valley for the relaxing walk downhill back to Klosters. There are many variations of this walk. A minibus to Alp Sardasca provides a more challenging walk towards Klosters (3 hrs). Or there is a 5hr walk from the Silvretta Hutte at the foot of the Silvretta glacier. A different minibus turns right at a fork in the valley to Berghaus Vereina. From there it is possible to walk to the 2600 metre Jorilepass before descending down yet another valley to Davos (8hrs).
A much milder day, taking the funicular up the Schatzalp. Explore the newly restored Alpine Gardens (1hr). Hiring a toboggan on wheels for a 1km descent is irresistible! This operates throughout the summer months and costs a purely nominal amount. Then walk through the forest back to Davos past very tame red squirrels (another 1hr). They will even eat nuts from your hand! This outing provides time to explore Davos itself before returning to Klosters. Even prams can mange the walk from Schatzalp to Davos.
Take the train to Davos Dorf and then walk along the Davoser See. Walk along the lake to take in the views of all the sailing boats and surfboarders. The water is so clear and still, that two views are to be had. One of the actual boats, and the other of perfect reflections! Then turn right towards Davos Laret at the far end of the lake and walk through the remote and isolated valley of Monchalbach back to Klosters (4hrs in all).
Sightseeing and Activities
For non-walkers, and those who don't even want to participate in all the other outdoor sports, there is still plenty to do and see.
The village of Klosters itself is small, but very attractive. A morning wandering around Klosters, visiting the local museum of arts and crafts is well worthwhile. From the beginning of June until October a ride up and down the Gotschna cable car, that starts opposite the railway station, rewards with fantastic views over mountain peaks. The nearby town of Davos is much larger and offers a wider range of shops as well as a Winter Sports Museum, a Toy Museum and an art museum dedicated to famous local artists. The15 minute train ride between Klosters and Davos is a well known tourist attraction in itself, with glorious views across the valley and the pine trees of forests almost brushing the train as it spirals around tight corners to gain and lose height.
A journey down the Prattigau valley. Explore traditional wooden villages, that make up the Prattigau valley that runs all the way from Klosters to Lanquart. The whole region is well known amongst the Swiss as a large recreation area. At the end of the morning, why not visit the vineyards at Malans, or just have lunch in the town? Spend the afternoon in Chur, the regional capital, as described in "Tourism/Excursions". Alternatively, turn right rather than left along the Rhine valley and visit the elegant spa village of Bad Ragaz. The hot mineral springs still provide the basis for a huge variety of treatments from massages to hot and cold saunas. Another 10 minutes' drive is the castle of Sargans. Perched on a hill, it provided a commanding strategic guard post across the main valleys from Germany and Austria. The Principality of Liechtenstein is on a hillside opposite Sargans.
The Engandine Valley lies a 20 minute train ride through the Vereina Tunnel. Trains go through it, either with passengers, or carrying cars, every hour from Klosters. The Engadine Valley is a wide, high plain just 3km across that runs for some 30km from the Austrian border to St Moritz. Very different in character from the Prattigau valley in which Klosters lies, the architecture, even the language is different (see "Excursions/Tourism"). A long, but worthwhile day out is to take the Bernina Express from St Moritz or Pontresina in the Engadine valley to Italy. The train ride is famed as one of the most spectacularly scenic in the world. Lunch in the small Italian town of Tirano, around 2 hours from the Engadine valley before setting off for the climb back to the Swiss valleys.
A day in Zurich, the largest city in Switzerland, or in Luzern, famous for its lake and medieval bridge that appears on so many post cards. Both are just two hours away by car or train.
Each of these days is enough of an adventure, that the holidaymaker will need at least one, or maybe two "rest" days to intersperse the travels!