The Painted Shader can basically do exactly what the (actually rather limited) tools in Terragen Classic (0.9) could do, and a lot more. To get started I suggest this simple workflow:
1: In the default scene, on the Terrain layout, at the top of the node list on the left, click Add Terrain -> Displacement Shader -> Displacement shader.
2: Open settings of Displacement shader, easiest by clicking the node in the list on the left and the settings come up underneath
3: Click the button with the green + and a small > arrow to the right of the "Function" field
4: Create New Shader -> Color Shader -> Painted Shader
5: Click the paint brush icon in the toolbar at top of the 3D Preview, select Start Painting Shader and then choose the shader you just created, probably "Painted Shader 01"
6: Now you're in painting mode.
I suggest opening the Painted Shader settings so you can have more control. From the default view it might be best to use a view-relative brush; this way it will have a small size close to the camera and a larger size further away. The reason I suggest this is you're close to the ground in the default scene so the large default brush size of 1000 (meters) will quickly cover the entire area close to the camera. It's often useful to have a fixed brush size, but for this quick demo do try view-relative I'd say.
7: Anyway, do a little painting now, click in the 3D preview and drag around (on the terrain; avoid painting on the sky

). You'll see white show up where you're painting.
If you stop painting for a moment and let the 3D preview refine, you might even notice there's a little height to it. That's because this Painted Shader is feeding into a Displacement Shader, causing displacement on the terrain in the shape of your paint strokes.
8: To get a more pronounced effect, go into the Displacement Shader settings and change Displacement Multiplier to something higher, like 10. Now look what you've got!
If it has gone as well as it has for me, you've got a terrain sculpted a bit in the foreground, and maybe even on the background mountains too.
9: Now let's say you aren't happy with some of what you did. Erase strokes! Click the Paint Brush icon again and choose Eraser Mode, then paint over some of the areas you covered before. Note how you're now removing "paint" instead of adding it.
10: For finer control, be sure to go back to the Painted Shader settings and look at Flow, Brush Falloff, and again the brush size parameters.
You can zoom out the camera, switch to an Absolute brush size, and then paint more controlled features. In the default scene I still find the default of 1000 meter brush size to be too big. Maybe 100 m is better. Or perhaps 200.
You also may find the 3D preview doesn't update quite as fast as you want for interactivity. This is something we're working on. In the meantime once you get a good sense of how the displacement will work, you can let the preview update to max detail (it will say "Finished Rendering" at the top of the 3D preview) and then Pause it, do some painting without having to wait for it to update, and then unpause it. It won't update while it's paused of course, but it lets you paint at max detail and does show your paint strokes in color (default to white, you can change the color though!), and then when you unpause you can see the displacement effect. It works better when you're painting for other purposes...
11: Be sure to click the Paint Brush icon one last time when you're done and select Stop Painting.
Now your preview will return to normal and you can see what you've accomplished. If all went well, you've got displacement painted on your terrain! You can do this with a flat planet, or paint detail or shapes onto any terrain. Use larger brushes to rough out large patches, then smaller ones for detail. And the best part is at the end you can apply additional displacement and effects to get rid of the "hand painted"/lumpy look. Also a good trick is to use lower Flow values with large brush sizes and Brush Falloff values to slowly build up terrain shapes and avoid the lumps by blending them together better.
Last but not least, as I mentioned you can use this for other things. You can paint masks for populations or surface maps, you can even paint color straight onto the terrain. Follow the same steps as above, except do it in the Shader layout, Add Layer -> Colour Shader -> Painted Shader, then change the color to Red or something. Be sure to select the right Painted Shader when you enter painting mode (if you followed the steps above linearly, you'll have two; the one in Surfaces is Painted Shader 02). Then paint the terrain red. Woo! Hehe.
I hope that's helpful.

- Oshyan