To quickly summarize my situation, I am in my 10th grade year of the Science Technology Engineering and Math (S.T.E.M.) program at my school, the most rigorous program offered by my school. Freshman year my GPA averaged near a 2.0 due to being overwhelmed by the transition to high school, and a few family emergencies. This year thus far my GPA has averaged a 3.0 on an upward trend that can be expected to continue rising in the exclusively AP and Honors courses I will take for the rest of high school. I have been building a strong portfolio of extra curriculum and achievements, including Tae Kwon Do classes, tutoring at another school, Model UN participation, participation in two county spelling bees, some letters of recommendation, a few of my finest essays (writing is one of my passions and possibly my strongest talent) and an invitation to the Congress of Future Science and Technology Leaders which I could not afford to attend. Princeton is my dream school with strong programs in English and writing, psychology, and astrophysics, fields which fascinate me and bring out the best of my abilities when I have the opportunity to study or work in them. From where I stand now, is there a possible path which I can go on that possibly ends in acceptance to Princeton? Any advice would be greatly appreciated, thanks in advance.
Based on your grades and achievements so far, I would say that admission to Princeton is very unlikely. There are plenty of other schools where you will have a strong chance, but your GPA is just not up to the expected level for an applicant to one of the most highly selective universities in the country. I don't know exactly how GPA is calculated at your school, but typically, students who are accepted Princeton will have close to a 4.0 unweighted GPA. Although you have improved your grades significantly (which is awesome), you're pretty much stuck with a maximum 3.0 average by the time you apply to college.
My advice for the next year is to develop a more serious focus on a specific extracurricular pursuit and shoot for excellent standardized test scores. Both of these things will make you an attractive applicant to selective schools even if your grades aren't in the elite range. Right now, you have a wide-ranging list of extracurriculars that are solid but not necessarily outstanding. Think about which is most important to you and how you can flesh it out into a larger and more unique accomplishment. This article has more details on what I mean (we call it "developing a spike").
Try not to be disheartened just because of your prospects at Princeton. All the program qualities you mention can be found at other colleges, and you may discover that another school fits better with your personality. Many students have a narrow view of their options for college because they focus on famous schools that they've heard are "the schools" to get into if they want to be successful. It's hard to get out from under these expectations, but trust me when I say it's the wrong way to look at things. Your college experience will be what you make of it. I actually wrote a whole guide on choosing the right college if you want to check it out as you think more critically about your options.