Exactly, UV coordinates (uvs…, beg your pardon) are created if you unwrap object. I saw you mentioning “repeated texture” - that happens when each face takes all UV coord space - You could do UV unwrap - command “Reset” and say for a default cube all faces lay on top of each other in such unwrap. As a result each face gets whole image assigned - you have repeating texture. Or your uv unwrap is sized bigger than grid space in UV Editor window - you also will get repeating texture. If your uv unwrap is in the grid space it covers image you supposedly created and paint - no repeating should occur.
I should have looked in your file which i did just now - your unwrap is scaled larger than grid space; that’s why you see texture repetition. After unwrap it’s a good idea hit menu UVs - Average Island Scale and UVs - Pack Islands wich automatically brings unwrap into the grid space.
As a result texture image you have will cover all object which you might or might not want… This time your image is 398x398 pixels and it is not tileable which means result will be very low resolution for a whole object and you saw already what happened when it was “tiled”. While it would work on a bathroom wall or flor (with appropriate image) this does not work on a landscape.
For a new start i’d suggest clean file, new object, make UV unwrap (doesn’t really matter what type of unwrap you use for a test plane)
and next, create new image in UV editor; choose size not less than 1024x1024 pixels. Your unwrap will automatically be in the image.
Now you can start to use Texture Paint mode.
By default image created is black; you could choose your own color - basis to start with. If you want you can use your initially used image as a texture for painting - Texture paint Stencil mode allows to position, scale and rotate image, if you brush over it ‘stencil image’ will be painted on before created texture image.
Do not forget to Save image you painted in UV editor window menu. It is not saved automatically when you save blend file.
For the tutorials - either Blender Cookie Members or Google “texturing landscape in blender”. One of my favorites is http://vimeo.com/5667694. It might look over a top at first and is not for latest blender version; nevertheless all needed is there; including a lot of other useful things.