Just 25km east of Tirana is Mt Dajti National Park. It is the most accessible mountain in the country, and many locals go there to escape the city rush and have a spit-roast lamb lunch. A sky-high, Austrian-made cable car, Dajti Express, takes 15 minutes to make the scenic trip (almost) to the top (1611m).
The area surrounding the upper cable-car station is a bit anticlimactic. There are some 'family-friendly' activities such as shooting things with toy machine guns and hitting things with swords, or you can ride about a patch of rubbish-strewn wasteland on a pony. Get past this lot and some abandoned buildings haunted by stray dogs, and things improve immeasurably. The mountainside is covered in lovely, shady beech and pine forests and a number of walking trails wend up and down the mountain. There are some very hit-and-miss trail markings, so it's wise to try and find a guide. Many of the hostels in town can arrange a day's hiking or mountainbiking up here.
If you just want to enjoy the view, there are grassy picnic spots just beyond the upper cable-car station, or try the lamb roast and spectacular views from the wide terrace of the cable-car station restaurant.
To get to the Dajti Express departure point, take the public bus from outside Tirana's clock tower to Porcelan (40 lekë). From here, it's a 1.5km walk uphill, or you can wait for a free bus transfer (departures every 30 minutes, five minutes). Taxis seem to charge what they want to the Dajti Express drop-off point, but the trip from Tirana should only cost 600 lekë. It's also possible to drive or cycle to the top.