if we were to send a mission to mars (unlimited budget WE WISH!) to explore different methods that would enable us to thicken the martian atmosphere, what different types of data should this mission capture and how could this help is in exploring potential methods to thicken the martian atmosphere.
This is an Earth Science site, not a Mars Science site. However, there will be a manned expedition to Mars within the next 25 years provided a world war doesn't throw a spanner in the works. They won't be exploring methods of thickening the Martian atmosphere, partly because it is impossible to thicken it to any useful degree, by which I mean to a degree which would enable astronauts to wander about in the open without wearing pressure suits.
Even if you could find a way to melt all the CO2 in the polar caps (mainly the south pole), there isn't enough to create an atmosphere as substantial as you would find at the top of Mount Everest. There might be enough to raise average temperatures by a few degrees, but what use would that be? The first astronauts will be doing the usual things. mainly geological exploration, particularly with a view to establishing whether there was ever any primitive, unicellular life on the planet.
Another task will be to find sites which might be suitable for establishing a permanent base, one of the main requirements being a good water supply. They might also experiment with ways of using local materials for building purposes, in preparation for future expeditions building a permanent base. The fact that liquid water can't exist in the open might be a problem when it comes to making cement. The Romans made an excellent cement out of volcanic tuff, so in the right place there could be plenty of that. Measuring the effects of the Martian environment on their bodies will take up much of the astronaut's time. They will not be exploring methods of geo-engineering.
We are already knowledgeable about some aspects of the requirements needed to terraform Mars & the Moon. As @David_Garcia_Bodego mentions in his comments, gravity is a major factor in celestial bodies being able to retain an atmosphere.
This is because the gravity of a celestial body affects the escape velocity of that body. The other important factor is the surface temperature of the celestial body. This is why Titan, a satellite of Saturn, has a thick atmosphere, despite having a similar escape velocity to the Moon (Luna), it is much colder and has a thick atmosphere.
Another factor that assists Titan in maintaining a thick atmosphere is its distance from the Sun and the weaker effect of the solar wind at that distance. Also, Titan is protected by Saturn's magnetosphere.
I recommend reading:
and the links contained within those questions.
Yes, these questions discuss the Moon and not Mars, but the information provided in those questions is applicable to Mars.
The graph below provides a visual of the gases a celestial body can retain in a atmosphere based on escape velocity and temperature.
Factors that act in favor of Mars being able to have a thick atmosphere are:
- Mars is farther away from the Sun than Earth & Luna so it experiences a weaker solar wind.
- The surface temperature of Mars is cold, which reduces the energy of atmospheric molecules.
Factors that act against Mars being able to have a thick atmosphere are:
- The gravity of Mars and subsequently the escape velocity of Mars is small.
- The lack of a magnetosphere to shield the atmosphere from solar wind.