If I want to make a communication satellite, like MarCO, what formulas do I need to calculate the perfect antenna? Is there a difference between the X-Band and the classic UHF? If the cubesat is destined for some saturn moon, does it have to have a larger antenna due to the interference of its other moons? How do I receive the antenna information?
There are too many general questions here that require book-chapter-length-like answers. I would like to specify first:
Do you want to calculate antenna performance, antenna pattern, or what exactly? Additionally, which antenna do you have in mind: ground station antenna for an uplink scenario, antenna on-board of the CubeSat for a downlink? In terms of performance, I would suggest to look for such calculation generalization as a link budget, as by calculating solely the antenna performance you won't be able to say if you can successfully close the link with your planned CubeSat. There are tons of examples in Internet and scientific publications, thus I advise you to review those first. For instance, this presentation will give you a good feeling what link budget is.
There is, of course, difference between bands. X-band is a frequency band between 7 to 11.2 GHz, while UHF is between 300 MHz and 1 GHz. In terms what frequency suits your mission in a best way, it is again quite broad question. It really breaks down to the mission requirements and subjected towards thorough trade-off analysis.
Larger antennas on CubeSats (and satellites in general) are nothing to do with interference. The main advantage of a larger antenna (let's assume parabolic reflector) is the larger gain. What is gain? See here.
You, of course, can do some engineering assumptions regarding your antenna when doing your link budget calculations. However, without prior knowledge of state-of-the-art or specific antenna device in mind, it is a hard task. Considering the desired deep-space scenario, I would recommend you to do a thorough literature research, check the flight heritage of successful missions, what kind of communication strategies they used. Having the previous experiences in mind, you can start an iteration process where you combine all the elements of your communication scenario into a link budget calc and see what kind of resources would you need to successfully close the link. This process would take a while, as it is considered to be a compromise between the desired performance, available power resources and current state of the technology.
But again, I advise you to make some effort and provide us with some rather more specific questions. So far there are so many ambiguities in your question that you cannot expect anyone would provide you with precise answer.
There is no such thing as a "perfect" antenna. Antennas are only better or worse than each other for a specific purpose. Choose a different purpose, and generally a different antenna will become best. Antennas embody design tradeoffs; having an outstanding strength in one application usually requires having outstanding weaknesses in other applications. Antennas with no notable weaknesses also have no notable strengths, unless lack of weaknesses is itself your goal.
The constraints of your mission clarify things considerably. Advice I would give for generic cubesats in low Earth orbit is not applicable when your target location is Saturn. Given that, you are pretty much forced into the highest-gain antenna you can manage, because Saturn is vastly farther away than typical satellite orbits, so you need to make up the extra free space propagation loss somehow.
The main problems you will have are size and control.
First, for a given transmission frequency, increasing antenna gain is most easily done by making the antenna physically larger. There are other things you can do, but they require complicated design details, like nonuniform aperture weighting. Physical size is a very tight constraint in a 1U cubesat, and you will be seriously pushing the limit of what you can carry, deploy, and control.
Second, the higher the gain, the more it suffers from being pointed incorrectly. Thus, the more important it becomes to both know where you need to point, and be able to stay pointed there, minimizing drift and jitter. For a LEO cubesat, I would advise going with Uwe's comment, and making the gain as small as possible, to remove or at least reduce precision pointing requirements, in order to get away with much simpler, smaller, and cheaper attitude determination and control. In your case, there's probably no way to avoid using a very high gain antenna, since the angular size of the Earth as viewed from Saturn is miniscule. You only need to track the planet, not a particular ground station, but the planet looks so small that the pointing requirements aren't all that different.