You'll need approx. 2 hours for this route that starts at Sestine - old village north of Zagreb. (Take bus no. 102 from Britanski trg). The colored roof of the church is hard to miss. Walk by the cemetery in direction of the Sestinski Lagvic restaurant, and then through the woods (on the left hand side from the road) on the marked path.
After about an hour you'll reach Medvedgrad - ruins of 13th century castle that offers one of the best views of Zagreb. A monument named "Homeland altar" was built here in mid-1990s.
Continue walking and soon you'll reach old Zrinski's mine - 17th century galenite mines, now unfortunately closed.
Your final destination on this route is Graficar mountain house - built on a small meadow as the alpine house. Nice restaurant on the ground floor and 32 beds in rooms above. You can walk down or continue uphill some 20 minutes until you reach the Upper station of the cable car.
This route also starts in Sestine, but this time take the marked path on the right hand side from the road after the Sestinski lagvic restaurant. You'll need a little bit more than 2 hours to reach Sljeme.
Kraljicin zdenac (Queen's Well) is the source of one of the streams, named after the legendary 15th Century Black Queen. Kraljicin zdenac is a nice place to rest, and there's also a small educational garden with explanations of natural phenomenons of Medvednica.
From Kraljicin Zdenac a route leads to the Graficar mountain house. This route was built in the 1980s by the "Pioneer Youths" so it's called "Youth's paths". It's one of the easiest paths climbing to Sljeme. You'll end up at Graficar again.
This is the path we used to take as part of the regular "sports activities" at the University, so I'm using it most often by pure inertia.
It starts at Dolje (last stop of the tram #15) - you don't pass the tunnel but climb sideby and take the path that starts above the tunnel. There will be dogs barking behind the fences at the start but soon you'll continue through the mountain path.
Adolfovac is the area named after Zagreb mayor Adolf Mosinski who did the road to Sljeme. Here you'll find a small pavillion named "Vila Propuh" (Draught Villa).
Continue walking upwards - this time you'll be at "Leustek's Path" that will cross the Sljeme road several times. On the upper part of the Leustek's Path you'll notice the plates explaining the types of the woods.
The path ends at Dom Zeljeznicara (Railwaymen's home) - a mountain house that's now closed. It is situated at the lowest end of the Cinovnicka livada (Clerk's Meadow) - a popular sledging area during the winter.
Don't miss the nearby Sljeme chapel situated on top of the Cinovnicka livada. It was built in 1932 by Juraj Denzler as the monument to 1,000 years of Croatian kingdom.
If you turn left from Cinovnicka livada you'll be at the Upper station of the Cable car in 15 minutes.
From Dolje till Cinovnicka livada you'll need approx. 2 hours.
This path also starts at Dolje (last stop of the tram #15), but this time pass through the tunnel an then continue walking by the Lower station of the cable car. Bliznec is the name of the small stream, but also of an old village mentioned as early as the 13th Century. In one of the nearby querries a Roman altar was found recently.
Take the marked path and in approximately 1,5 hours you should end up at Puntijarka. It is the mountain house (also known as the "Ivan Packovski" mt. house) with an excellent restaurant and 35 beds in rooms above.
If you want to take the cable car on your way back you'll need approx. 30 minutes walk on top of Sljeme.
Favorite thing: This path was recently made and also starts at Bliznec (close to the Lower station of the Cable car), but goes along the Bliznec stream. I didn't take it yet since it is paved and looks very artificial, but on the other hand it is the only path that's wheelchair-accessible and there are also some educational posts along the path so it's probably worth a try.