Kaleybar is in Iranian Azerbaijan.
Kaleybar is usually pronounced as "Chaleyber" (with Ch pronounced like in Chief). For German speakers, pronunciation sounds like "Tschalibär".
There is a bus leaving from the Tabriz Central Terminal (South of the city) daily at 7am. The company is Mahan Safar and it costs 100,000 IRR.
The 2-3 hour bus ride offers dramatically different landscapes of greenery and mountains. The bus drops you off along the main street opposite a gas station.
Impressing Babak castle . A taxi to the carpark behind Babak Hotel (about 7km from Kaleybar town) in the mountains is about 2$ / 80.000 IRR (Jun 2016) from Kaleybar. From the carpark follow the obvious path to the castle (you cannot see the castle from there). If you are fast, going up to the castle takes about 75 min. Return trip to the carpark usually about 2.5h. From there hitchhike back to Kaleybar or call a taxi.
There are decent homestays around Kaleybar town charging around 10 USD a night. Most hosts do not speak English, so communicate through Google translate or find a host through couchsurfing.
There are savaris (shared taxis for 4 passengers) to Ahar (about 1h, about 2$ / 80.000 IRR p.p. when full, Jun 2016), leaving when full.
From Ahar you can connect to Ardabil by first taking a savari to Meshgin Shahr (about 1h, about 2$ / 70.000 IRR p.p. when full, Jun 2016). There you may have to take a (shared) city taxi for the Ardabil terminal in Meshgin Shahr (just a street crossing with a few waiting savaris). From there you can take a savari to Ardabil (about 1h, about 3$ / 110.000 IRR p.p. when full, Jun 2016).