As far as what contexts to use and what to use them for, this is a personal decision. If you search this board you will see a few threads in which people list the contexts that work for them.
I know of no better way for you to get this done than to start with the default contexts and see if there are some you never use (or forget to look at) - delete them. Maybe you will see the need to add a new context. Whatever helps you get things done is the right thing to do. For example, I don't use the @anywhere context, and I put all my work-related calls in @office not @calls - works for me!
Re: starting a project...
I often start with the tasks. I create the new project, then brainstorm a list of tasks. I put the list in the note section of the project master task if I think that the actions will be quite far in the future (months). I see them at each weekly review and can create active next actions from them as needed. For actions that are really next, I create outlook tasks immediately.