Section 106 of the Copyright Act explicitly defines what your rights are as an owner of copyright. Unless otherwise stated, you have the right to:
It’s important to note you also have the right to not do these things and prevent others from exercising these actions without your express permission.
You also have the right to give these rights to others. For example, you may give your publisher the right to reprint your work and distribute copies under certain conditions. You can also give performers the right to perform this work publicly.
Be sure to read any and all contracts regarding your copyrighted works carefully! Preferably, have a lawyer read through it and make sure you retain any of these rights that you would like to. This also applies to grants. Sometimes, by receiving a grant to produce a work, you may also be giving up certain of these rights or may be required by contract to exercise these rights in a certain way (such as making the work publicly available online for free)
Arrangements are considered derivative works, which means that they are based off another, original work. This also applies to new editions of books or scores, translations, and movie adaptations. This Guide made by the U.S. Copyright Office says that in order for an arrangement to be copyrightable, "a derivative work must incorporate some or all of a preexisting “work” and add new original copyrightable authorship to that work."
As an arranger, the only parts that are copyrightable are the additions you make to the original work. So you don't own the original parts of the music, but the way you arranged the piece and any/all changes you made to it are owned by you. You can still register a copyright on an arrangements, but only on the parts you created.
You do still have to be licensed to make a derivative work. Authors of copyrighted material have the exclusive right to authorize and create derivatives, so if you want to arrange a work, you need to get permission from the original creator first. This can usually be done by contacting the composer or whoever manages their estate. You may be asked to sign a license and pay a fee and will likely need to pay royalties based on your agreement with the original composer.
It is not necessary to register your copyright with the Copyright Office, since copyright is automatic once you’ve put your music down on paper. However, this may be a good thing to do anyways. For example, in order to file a federal lawsuit alleging copyright infringement or to receive statutory damages, you should have a registered copyright. It’s also just good to show the world that your work is copyrighted and that you own that copyright.
All registration for copyrights can be done through the Copyright Office.
Note that this does cost some money (Likely around $40 but you can see the fee schedule here) but could be worth it in the end. One thing to note is that it is a lot cheaper to register online.
You are the creator of your work unless you agree otherwise. You may be hired on commission to write music for a company or a media project or asked to compose something while working as an employee of a company. What this means is that you are paid for the fee of your work (either through a salary if you are a regular employee or in a commission fee for commissioned works), but you are also giving up your rights as creator. This is not necessarily a bad thing, but is crucial to know for future projects you may undertake so that you don’t lose out on any royalties you might expect to get.