Can we seen an example of what you're talking about?
The lines could be interlacing. They could be compression artifacts. Post a screenshot of your project with the footage selected in the Project panel so we can see the settings, the footage in a timeline with all modified properties of the layer revealed (press the u key twice) and the Magnification factor set to 100% and the resolution set to Auto, then move the comp panel around so we can see the problem areas. Print screen if you are on a PC then paste into the reply field on this forum or Shift + Ctrl/Cmnd + 3 on a Mac to save a screenshot to your desktop and drag it to the reply field so we can see what's going on.
If it's interlacing you can fix the problem a bit by properly separating fields. If it is compression artifacts then you are out of luck.
I remember you I'm a super newbie. Adobe AE, in this case, is italian. Res is 1280x720p, framerate 25fps, AR (16:9), but there are a lot of horizontal lines. This horizontal lines are present only during a fast motion scene (static or low motion scenes doesn't present lines).
However, how can I separate the fields?
If that truly is progressive footage, somebody along the way baked interlacing into it and you may be a bit screwed.
However, try looking in the interpret footage dialog and choosing upper field first instead of progressive and see if that helps.
Open Interpret Footage from the Project Panel with the footage selected. Check Separate Fields...
Without third party software that's the best you are going to get. You can check for proper field interpretation by temporarily doubling the frame rate of the comp and making sure that the motion is all in the same direction. If you have fields reversed then one frame will have motion going in one direction and the next frame going in the other.
You can decide to work at double the frame rate (actually gives you a little more data to work with) and then drop the completed comp in a new standard comp for rendering making sure to turn on Collapse Transformations and pixel motion frame blending to get your best results.