Without knowing your exact budget, and exactly what you are wanting to accomplish, I'm going out on a limb here and going to assume a couple of things.
You're a young person, probably
a student and living at home and money is a big factor that you don't have a lot of. You're wanting to post to Youtube, but really aren't in to it to record the next big hit. Piano and flute is pretty much what you expect and want to do. You at the moment are using a webcam to produce your videos. Is that about it? Assuming these assumptions are correct and also assuming you may wish to purchase a better camera down the road. First thing first. Let's work with what you have, and as your ability and knowledge grows so can your equipment list.
I wouldn't even worry about a dedicated audio interface at this point. If it were me, I would get a USB power condenser microphone. Something like the Behringer C-1U.
Behringer: STUDIO CONDENSER MICROPHONE C-1U
You can hook it right up to your computer, and it should record the audio right in to your webcam feed.
Get a mic stand with a heavy base to support your new mic.
You'll want to place your mic just above head high and about 2 to 3 feet away. That would be half to just less than a meter if you are on the metric system. You want to point the mic down towards your flute about half way down the instrument. The flute is a cool instrument and the sound comes from the entire length so you want to be able to capture not only what is coming off the mouth piece, but from the holes as well. The flute is also a very dynamic instrument and you're going to get notes louder than others at times, but experimenting and experience will teach you a lot on where and how to position the mic to get the best result under your conditions and in your untreated room.
Later down the road, if and when you want more
professional results, then I'd worry about getting an audio interface, a more expensive camera, good editing software, better microphones etc. In the meantime, hang out around here and forums like this one, and even ones that lean more towards video. Soak up the knowledge that is freely available. Learn the ins and outs. What mics do what, what interfaces do what. What is SMTPE timecode and how does it relate to video, where do I place my mics etc. etc. Then when you are more informed you can make better calls on the path you wish to
travel.
With that said, welcome to the forums, happy recording and good luck to you. One last thing, don't be afraid to ask your questions. The only true dumb questions are the ones you refuse to ask.