When creating a new task it defaults to the Inbox, as this project is set to be the quick way of adding tasks, for when you just need to quickly add a task, and edit it later. It being the default can not be changed, though when creating a task you can quickly select another project.
Though, Today is a view and not a project. If you are referring to your startpage, when you first load the app being Today. This is synced from the web version. At Todoist.com, top right corner Settings, from there you can set your start page.
Hey Shirley ... I know this is a VERY late response and you probably figured it out from his question but just in case (seeing as I got here due to the same question):
Yes, if you change the default Start Page on the web or desktop version, it'll change it for the app allowing you to just hit that lovely "+" sign and have the new task "default" to your inbox vs. the Today or wherever your start page is.
Thaks so much!
You are most welcome!