Dark spots on the first photo do not appear on the second, so it looks like I cleaned the sensor from the dust, but there new are light spots of different colors on the second photograph. How do I get rid of those? I repeated the process two more times, but still have those light spots.
Spots caused by dirt and dust on the sensor are dark, by nature of being shadows - your second shot shows lens flare, which is caused by reflections within the lens - it may be a side effect of your sensor now being sparkly clean (after all, digital sensors are far more reflective than film ever was), or it may be that there is a little bit of dust within the lens - it just be that your shot is towards a bright light source, which is always a high-risk for flare.
If you're still worried about dust, the recommendation would be to shoot a known plain surface, like a sheet of paper, rather than the sky.
The dark spots are dust. You managed to clean then. Good job!
The light spots are not dust, they are flare. That is caused because by sunlight entering the lens. Even though it is out of the frame, some stray rays can enter.
First, you can easily confirm this by shooting away from the sun.
Then, you can learn to minimize flare by using a lens hood. Lens hoods are not 100% effective and almost always (there are exceptions) become less effective when you zoom in.
If it fails, you will simply have to point your camera in another direction or wait for the sun to move.
If you are talking about the circular colour spots in the top left of the second image, these are not caused by the sensor but rather are caused by the sun hitting the front of your lens at an angle. Try photographing an image at 180 degrees from the sun to ensure you do not get lens flare.
When cleaning your sensor a better method to see how clean it is would be to photograph a well lit blank white sheet of paper, using your highest F-stop (such as F22) and out of focus ... any foreign objects, smears or stains on your sensor or lens will then show either fully in focus or very nearly in focus. Personally I shoot a blank wall at home on F22 and move the camera while the shutter is open, then I know any impurity in the image has to be on the sensor or lens.