How to check if the gear shifter or the gear itself is broken? I'm not a bike expert so I'm not sure if I'm using the proper terminology here. So I bought a used city bike. It's the first time I see such gear and such a bike actually. It looks almost like this bike: http://www.fahrrad.de/kreidler-vp2-schwarz-glanz-385180.html

However there is only a single issue with the bike. The speed switcher only changes from speed 1 to speed 2. I can't change to higher speeds and I don't know what the problem is. When I try to switch to 3 for example the switcher doesn't allow me to, so maximum is speed 2. I went to a bike shop, the guy seemed to be not knowledgeable at all. He said the problem could be from the switcher itself and it would cost 10 Euros to replace it, but it could also be the entire gear is broken and that would cost 200 euros!!! It felt that he wanted to ripe me so he wasn't really willing to tell me what the real problem is.

My question is: how to find out which part is broken?

Here are the images for the switcher and the gear itself. And I would highly appreciate it if someone knew what the issue might be so he can also mention the name of the broken part so I can order it online!
This does not answer your question, but it's too long for comments.

Summary: take it to another bike shop.

Details

This is an internally geared hub (IGH). I cannot tell what make it is; the ones I know of are Sturmey Archer, Shimano Nexus and Alfine, and SRAM.

The whole bike has been left to molder - the chain has rust, and the tires are flat. The chain should be fine with some oil.

The question is - why was it left to molder, and why was it sold? My guess is because of the gear problem.

Advice

See if you can identify the brand or make. Someone might be able to help more if you can tell them that. Once you know that, you can also look for a bike shop that knows about those particular hubs.

IGH's can stop working for fairly trivial reasons, and often the new parts are not very expensive. The main cost is the mechanic's time.

While you could DIY, the fact that you asked the question suggests that that's not a realistic option for you. There is no guarantee that after taking your first IGH apart you'll be able to put it together again. There are lots of springs and pawls and bearings and gears in there ...