To create a new project, start by selecting a task and going into edit mode. On the right side you will find the project picker. To add a new project, simply place your cursor in the text field. Type your project name and click the add button. Notice how the project is automatically selected for you. Context's and tags are added the same way.
There are slight differences in the way tags are handled versus context and projects. While context and projects can have only one item selected, tags can have many items selected. So as you add new tags they each become selected and thus applied to the task being edited.