I've been exploring the world and I've found some probe sites and some item stashes that require a higher Mechanical field skill level, but I'm not sure how to level it up or any of the other field skills for that matter. How do you level up field skills?
In order to level up your Field Skills, you need to be a BLADE operative. You will eventually become one as the story progresses.
From there on, with every BLADE level you gain, you will need to talk to Eleonora at the BLADE Concourse in the Administrative District in order to gain a single level of whichever Field Skill you wish.
It is recommended to level the Mechanical Field Skill before the others, as you'll need it to place Data Probes, which are a very important source of Miranium and Funds, and also serve as Fast Travel points.
Each Field Skill can be levelled up to level 4 at first. In order to reach level 5, you will need to complete the mission "Off the Record" from the BLADE terminal for each Field Skill that has already reached level 4.
Since the mission appears and disappears at random, it is recommended you accept it as soon as you see it to prevent it from disappearing. Otherwise, you might have to visit the BLADE terminal several times before the mission reappears.