I downloaded the Salesforce1 mobile app and can't find how I to add a new contact using the app. Is this possible or is the mobile app functionality limited to only viewing data?
Has your user got permissions to create contacts? If so you should be able to create contacts from Salesforce1 if no you won't see the "New" button and you wont be able to create contacts from Salesforce1 and either from desktop-web browser. Check your user, the profile and the permission sets that it has. Hope it helps! :)