I got a 22 on the ACT without studying as a freshman. How high can I raise my score?

       I'm a freshman, and, unknown to me up until the day of the test, my high school paid for a mock ACT made by the College Board(I think?) and my score came out to be a 22, with no studying or any idea of what the ACT was and how it was formatted.  Our teacher started administering our test late, so we only had five minutes for the writing section, which I only managed to score a seven on.  With studying and a bit of preparation, how high do you think I could raise my score.  I'm not necessarily shooting for the Ivies, although my dad did get into all of them, started at Harvard, and then transferred into MIT,--I just want to keep my options open.
      For a general idea of my work habits, my unweighted GPA is a 3.57. We have seven classes, and right now I'm taking:
     Honors U.S. History--A+ for both quarters and exam, A+ for 1st semester 

(AP was an option, but time was a problem, but I’m taking both AP Human and NSL next year). I got an A both quarters and exam.
Piano 1a–A+ for both quarters and exam, A+ for 1st semester
Honors English 9A-- A+ for both quarters and exam, A+ for 1st semester
Honors Chemistry–B+both semesters, B+exam, B+ for 1st semester
Spanish 3–B+ both semesters, B+exam, B+for 1st semester
Honors Geometry–B both semesters, B exam, B for first semester
P.E–A+ for both quarters, A+ semester
So, do you think that I have a good chance of raising my act score to be in the 32-34 range? And, as a side note, do you think that my schedule is rigorous? Would I be able to handle AP NSL, AP Human Geo, Foundations of Technology, AP English Lang, Honors Bio, Honors Spanish 4, and Honors Algebra II next year?

Your score will automatically improve by you doing your regular school work. When it comes time to take the test for real you should study for it. The ACT is a time crunch exam. If you can speed read it will help you a lot.

What are you component scores? Math will go up simply by completing Algrebra II and Geometry; better if you take trig and get the basics of prob and stats. Science will as well. Reading: if you are not a reader, you need to build your vocab. Take another practice test second semester sophomore year or summer after sophomore year and then start looking at what you missed and shore up from there.