First, confirm a check of menu - 'View' - 'Show textures'. If it isn't checked, you check it.
Open a material panel from menu - 'Panel' - 'Material Panel', and push a New button on the material panel for creating a new material.
Push a 'Prop' button on the material panel, and specify am image file of 'Texture' mapping. Push a 'Ref' button and select the existing image file, or input a file name (including file extension) directly for creating a new image.
Put a plane or a sphere with [Primitive] command. Then, a texture mapping is wrapped like a right figure. If a texture mapping isn't wrapped, select faces with menu - Edit - 'Select all faces' and call menu - 'Selected' - 'set material to faces'.
When you use not [Primitive] command but [Create] command, it isn't assigned the UV mapping. You must assign it with [UV] command.
When you select [Paint] command, a paint panel is opened automatically. Then, click either on the 3D view or on the paint panel, and it displays that confirm to begin painting. You select 'Yes' and it's all right. You can paint a mapping image both on the 3D view and on the paint panel.
Call menu - File - Save in the paint panel, and a mapping image is saved to a disk. Be careful that it doesn't confirm overwriting a file.
See a FAQ if you trouble that a texture mapping isn't displayed.