Buses from Istanbul take about 6 hours and cost 45-50 TL. Buses from Ankara take around 3 hours and cost around 25 TL. If you want to go by car from Ankara, it is 220 kms.
Dolmuses regularly drive the 2 km between the old town (Çarşı) and the new town center, which is where buses are likely to let you off / pick you up. Once in the old town, everything is reachable by foot. Additionally, some intercity buses might drop you off in the nearby steel down of Karabük. From there you'll have to find a dolmus into Safranbolu.
Safranbolu is famous for its Ottoman houses both in Çarşı (old town) and the neighbourhood of Bağlar (literally "orchards"), and qualifies as a UNESCO World Heritage sight.
Mencilis (Bulak) Cave which is 3 km from city center, worth to see since it is the 4th biggest cave in Turkey.
Take a look (and a tea?) inside the Cinci Hanı, a massive hotel in the center of the old town. Wandering around the city you will take in much, but to see inside the houses, head to one of the historic homes open to the public. Kaymakamlar Müze Evi is located just above Bastoncu Pansiyon, to the south-east of Cinci Hanı. Keep walking up the hill, and take in some beautiful views at Hıdırlık Tepesi.
Traditional table clothes and souvenirs from Arasta Bazaar.
From Safranbolu you can go, with bus or minibus which you can find at the bus terminal, to Amasra, a little touristical Black Sea village, two hours away, which is popular with local people as well as with turkish tourists coming from Ankara and Istanbul. The Castle of Amasra and Bird's Rock Road Monument, herited from Romain Empire, are the two main historical places that should be seen. The beaches are crowded, so you may choice to swimm at the places near to the castle or the breakwater if you have confidence in your swimming skills. In Amasra it is also possible to find some good fish restaurants with a good view of Karadeniz (Black Sea).
7 km away from Safranbolu, Bulak Village is a well preserved little village which can be reached in 15 minutes from Safranbolu. Unfortunately there is not enough means of transport as like as Safranbolu City Center. Taking the "dolmus" for Karabuk, than changing it for another dolmus to Bulak, at the very beginning of the little and deformed way to Bulak at the right hand (which is the midpoint of the main avenue between Safranbolu and Karabuk) is the most practical option. Bulak Village gives his name to the Mencilis Cave, the locals call it "Bulak Mağarası", which is one of the most deeper caves of Turkey. It is possible to reach Bulak village from north, following the Mencilis Cave Route, without turning back to Safranbolu, instead of that going forward to south, following the way ahead between the cave and the Village will lead the visitor to Bulak. That is a short, but worth to explore trekking parcours also. Old houses, alleys paved with stones and the little hamam of the village make from Bulak an outgoing attraction point near to Safranbolu, which is not explored enough yet by the tourists.
8 km away from Safranbolu, Karabük is the first industrial city of Turkish Republic, with his Iron and Steel Factory (Kardemir A.Ş.), constructed by Soviet technology in 1937. As a young industrial city it is 2,5 times bigger and populated than Safranbolu (which is administratively a sub-district of Karabük), but touristically there are not too much things to see, but if you are interested in seeing how the people of this industrial city lives it is worth to try; also the outskirts of Karabük elarged to the hills at the backyard has little and sympathic village-looking streets populated by the workers family (like a turkish "How Was Green My Valley" scenery).