Welcome. First, I would suggest reading this excellent essay on the Peabody admissions site, about the different ways to study music and the different types of students best suited to each:
http://www.peabody.jhu.edu/conservatory/admissions/tips/doubledegree.html
You can major in music with a BM (conservatory/music school), which would mean 2/3-3/4 of your classes would be music (theory, solfege, history, technology etc.) and you would spend a lot of time in lessons, rehearsals and performing. These programs have auditions for admission, usually with prescreens.
You can major in music with a BA (college/university), which would mean, generally speaking, 1/4-1/3 of your classes in music (at Harvard it is 1/2) with academic music classes in theory, composition, music history, ethnomusicology, musicology and technology. It is actually possible to major in music and never touch an instrument! Some BA programs have performance elements, some don’t. Many or even most don’t have auditions. Instead, if you are talented, you submit an art supplement with music resume, recordings, letters of recommendation from music teachers, awards and so on added on to the Common App.
You can also double major (yes, tough with some subjects with exact sequences of courses or with labs that might conflict with music obligations), major and minor, or do a double degree. Oberlin, Bard and Lawrence are often mentioned for double degrees but there are many others, including Peabody/JH, Tufts/NEC (BA/BM), and Harvard (BA/MM). Bard actually requires all conservatory students to do a double degree.
Some of the most talented musicians at a school like Harvard don’t major in music at all, but study privately and perform in extracurriculars or professionally off campus.
So you see, there are many ways to do this.
I will tell you that studying music can be a fine preparation for careers in many fields and gives access to the jobs and grad schools (including med, law and business) that any other bachelor’s degree gives. I firmly believe- and I have three kids now in mid-to late 20’s, one a PhD music student- that if you follow what you love, things work out. You might love math more. You still have time to think about it. I bought a book called “The Musician’s Journey: crafting your career vision and plan”. The music world has changed with Internet, “new music”, freelancing and a lot more entrepreneurial activity.
It is fine to apply to both colleges and conservatories and decide at the end of senior year. My own daughter did that. if money is an issue, that approach is a good one. To me, very superficially, you sound like someone who might want to go to a college or university and either do a double degree or do music outside of academics. But that is just an initial impression. Read the Peabody essay and see where you think you might fit! And come back here anytime over the next year- people here are really helpful.