I'm trying to learn to play trumpet on my own (is this feasible?). I've never played instruments before. I'd like to know any recommendation for books, videos, and other resources you have in mind (it would be best if they are free or inexpensive) ;)
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Yes, it's definitely feasible. In fact, if you get some good beginner resources to start with, and practice enough, it's even easy. It's probably worth your while having at least a few lesson with a tutor after you've been playing a month or two, to make sure you're not forming any bad habits.
As for resources, get something that starts of very basic, with lots of tunes that you already know. Personally I've found the "Tune a Day for Trumpet" books to be quite good (both old and new), as they have lots of tunes you would probably have heard before, and they start of really easy, with good descriptions of how to play.
If you want to learn lots of technical detail on the instrument, and how to play it, check out Sound the Trumpet: How to Blow Your Own Horn, by Don Young. It's a very dense, but very readable book that covers the world of trumpet fairly exhaustively. It doesn't have many songs to play, it's more of an "about the trumpet" kind of book, so you'll need other resources if you want to learn songs.