We will spend a week in Tokyo in april and I am now looking into the best way to get around the city. The subway seems to go everywhere but there are also normal trains connecting most places. How does a JR pass compare to subway prices? Also - is it likely that we will use the subway anyway for some destinations? We will stay close to Ikeburo station which makes it easy to go by train. We may also do one or two day trips out of Tokyo by train. Mount Fuji or something like that.
I doubt that a JR pass will be worth it for just travel within Tokyo; usually with most rail passes, you need to be making several long-distance trips for it to break even. JR pass would make sense if you planned taking the shinkansen to/from Tokyo, and then you can consider the rail travel within Tokyo to be a "free" benefit.
According to this site, a 7-day pass costs about $350 - or $50 a day. Highly unlikely you'll be doing so much subway travel that you'll come even close to that!
Your best bet is perhaps to get a Suica card: if you arrive via Narita, you can get a Suica card as part of a ticket combination from the airport to Tokyo - called "Suica + N'Ex". You can then use the Suica card to pay for subways and trains around Tokyo, just tap it at the reader on the gates as you enter the stations. It comes preloaded with a certain amount, and you can top-up as needed.
Also, a quick note on how subway and trains work together: they are really two systems that complement one another; you'll likely use both. For example, the simplest way to get from Tokyo Station to Shinkjuku is by the JR Chuo line - especially the orange line, which is an express train that makes just a couple of stops between them. By subway, you'd have to make several transfers for this same trip. For some trips it makes sense to use a combination: a JR like to get you to the right part of town, then a subway for a few extra stops.
A JR pass won’t save you money on short journeys. Just pay the normal ticket price: it’s not expensive. To find the best route, use an on-line route planner. I recommend jorudan.
The larger stations are crowded and confusing, so allow extra time for connections. Be aware that sometimes there are two stations of the same name, but belonging to different networks, next to one other. (Or, in one case, one is under the other. It took me fifteen minutes to figure out where my connection was.)