You might, for example, be thinking of majoring in Business or Economics, which require at least the Calculus course MATH 220, and perhaps MATH 221. (MATH 220-221 is a "terminal" sequence--you learn concepts of calculus, but you are not trained at a technical level, so these courses would not prepare you to follow up with a course such as Differential Equations (MATH 246), Linear Algebra (MATH 240 or 461) or a calculus-based probability and statistics (STAT 400).
For MATH 220, the math placement test becomes more ... interesting. A preprequisite course to MATH 220 is the course MATH 113 (College Algbebra with Applications), or a suitable score on the Math Placement Exam. The material of MATH 113 can be covered in high school, but MATH 113 does carry University credit; you can see how you fare on this material by looking at the course syllabus for MATH 113, and perhaps checking out some past MATH 113 exams on TESTBANK, our online archive of past exams.
The advice here would be the same as above, with some additions. First, if you can get a good preparation in the precalculus material in high school--do it!--so long as you still get a solid algebra preparation. Every course opportunity in college is precious (not to mention expensive), and it is better to do in high school what you can.
Second, be careful about your choice of calculus sequence. MATH 220 is the right choice for most business or econ majors, but it imposes limits. For example, it is unwise (perhaps suicidal) to attempt graduate school in Economics after advancing in math only through the level of first year calculus. (Former Math/Econ double majors tell us that the math classes most useful to them in Economics grad school were their junior level theory-and-proof courses, Math 410 and 411.)