What bike would be right for me? I live in an area with many steep hills which are all about 1000-1300 feet high. My current bike is a Ghost Northshore which is a freerider just slightly under 37 lbs. I bought this bike to ride on my own trail. However I dont do this kind of riding anymore and I am looking for a bike to cruise hills up and down and to do tours under 20 miles regularly. Climbing with my Ghost bike is certainly not the funniest thing to do. Sometimes it is just not possible. The bike is heavy, has only higher gears, the suspension takes all my energy (it is not lockable) and the saddle is very low. Its just not made for this kind of riding. 

Now Im looking for a lighter bike which fits my needs a little bit better but I dont want to loose all the characteristics of my freerider. I would love to have bigger brakes and some fork travel. I found this Cube Stereo 140 HPC Race which has the right amount of suspension travel for me (I would like to take smaller jumps under 3 feet or so very rarely); bigger brakes; is light enough and most important: It is in my price range. 

I need your advice if this bike is capable for regularly and sometimes serious climbing. The weight seems to be fine but Im especially unsure if the gears are right for my purpose. Are they low enough for serious climbing? Or is this bike meant for riding in a bike park? That is something I almost never do.
There are far too many options and it's far too subjective for someone to give you a straight "get this bike" answer. 

You're kind of talking about the "All Mountain" to "Enduro" range of bikes. Ie bikes that can go long distance but can also go downhill. 

The all mountain type bikes typically start at 140mm-ish travel. The cube you link is one of these.

The enduro bikes typically start at 160mm travel. Cube do the 160 version of that bike too. 

The difference is, less travel, better pedalling efficiency, the more travel, better downhill efficiency. 

You just have to decide what is more important to you, having a nicer time going uphill and sacrifise some of the down (get an all mountain). Or just be good enough to go uphill, but the downhill is the main point (get an enduro).

For decent bike, like your cube, the difference between the 140 and the 160 will be marginal in terms of pedalling and downhill. 

You just have to decide, on which side of that margin you want to be.