People don’t complete the core before doing anything else. Completing the core happens alongside other classes.
The core really has two components – the part that has to be completed by taking special core courses (and that you can’t get out of with AP or examination credits), and the part that can be completed with courses you might take anyway (although in some cases there are special core-only options) and for which AP or examination credits can suffice.
The first part represents 9 quarter courses. The second part is 6 quarter courses if you test out of the language requirement entirely, and up to 9 if you don’t. For perspective, to complete a bachelor’s degree at Chicago, you need a minimum of 42 quarter courses, or an average of 10.5 per year, but most students wind up taking more than the minimum, so 11-12 courses per year (in three quarters) is pretty standard.
If you took nothing but core courses, it would take 3-5 quarters (out of 12 total) to complete the core, depending on the amount of AP/examination credit you had. Students usually try to get most of the core out of the way in their first two years (6 quarters), especially since (a) 2-3 Humanities courses must be taken first year, (b) the 3 Social Science courses are really integral to understanding what the University of Chicago is about, and © the courses in the second set are often required for their majors anyway. But it’s not uncommon to leave 2-3 of the Civilization or Arts courses for the third or fourth years, or to pack them into a quarter abroad second or third year. As a practical matter, that means that, over the first two years (6 quarters) on the Chicago campus, you take 5-10 courses out of 22-24 that you wouldn’t take anyway, plus another 3 if you have no language credit and wouldn’t otherwise take a language.
For someone interested in linguistics and computational neuroscience, every one of the second set of core requirements would be satisfied by something you need to meet the requirements for your major and minor anyway. For linguistics, you need to take (or test out of) three quarters of a non-Indo-European language, which would satisfy the core language requirement. For computational neuroscience, you need to meet the general education requirements for biology, which are completely consistent with the math, biological science, and physical science elements of the core. So essentially, for you, the “core” would mean devoting about 1/4th of your classes in your first two years to things not directly required for your major and minor, plus one extra (generally enjoyable) class every other quarter your last two years, or a quarter abroad.
You could try to self-study AP classes, or test out of them, but that probably wouldn’t be advisable based on your interests, unless maybe you could get credit for a year of Chinese or some other qualifying language. Self-studying math, biology, or chemistry could get you out of requirements, but could put you at a real disadvantage in the upper-level classes you need to take.