Our product is a cloud based application available cross platform. It's functionality, summed up, is to help you share the books you read with your friends.
We are in the stages of market research, but I personally do not know where to start my desk research. I know I can look at our competition on app stores, but to be honest I am not sure where to go next.
The main function you are offering is social by nature. Users can already do that using generic social network (e.g. facebook). I would start with two things:
The first part will give you a great insight on how to focus your product and the second part will allow you to start assessing how big the market is. You can probably say that people will be more likely to share on a dedicated network or with a dedicated tool but it's probably a 2x rather than a 10x of what people are already doing today.
Another great way to find out more about your market is by searching forum posts and replies. As comment above your business is based on customers being social and talkative with friends online. These people will probably be social on other networking sites like forums.
Further to this, why not ask your own quesitons on these forums and find out the answer to your own specific questions.
I would also note that before you start your market research you need to really understand who your customer is and where you are most likely to find them.
I'd recommend to hire professionals for several hours/days and be sure in the results. For ex you may check this guys: singularex.com