In Project Server permission mode beside granting permissions in SharePoint root site, you should give permissions to the users required for Project Server Access (for example, via assigning them into projects as resources, adding their Active Directory groups to Project Server groups, assigning their Project Server groups to Project Server categories, etc.). Project Server permission is a rather complex theme itself.
Further info to the theme:
Video series: How security permissions work in Project Server
Manage users, groups, and categories in Project Server 2013
Plan groups, categories, and RBS in Project Server 2013
Regarding the SharePoint permissions, the user need (as far as I remember) at least Read permissions set on the following sites / libraries (each of this entities have their own permission set defined):
PWA
PWA/Project Detail Pages
PWA/Project Server Workflow History
PWA/Project Server Workflow Tasks
PWA/ProjectBICenter