Can I transfer music from my old iPhone to my current one?

Like if I sign into my apple account would I be able to put them on my new phone kinda like how the iCloud works

Yes. When you get your new iPhone, go into iTunes, then go to "more" all the way to the right on the bottom, then go to "purchased" then go to "music" and then tap "not on this iPhone" and you can download any song you want onto your new phone.

You can transfer music from your old iPhone to current one in may ways. The simplest way is to try some mobile transfer programs. Download and run such applications and make your devices connected to computer, then all your iPhone information will appeared on this program. Click file that you wanna it to be transferred to another devices and hit "OK" button. And your old iPhone files will be transferred to current one automatically in few minutes.