Flute fingerings need to be learned properly from the onset to avoid developing bad habits that are difficult to break later.
The flute has a three-octave range, although most commonly flutists, especially beginner to intermediate level players, use a two-octave section in the middle which includes the top half of the bottom octave to the bottom half of the top octave. Later you'll learn about some additional, higher flute notes.
An important guideline for flute fingerings is that each C, D, E, etc., has its own fingering. Don't make the mistake of using the same flute fingerings for each note in all three octaves.
Let's start with C flute fingerings. There are actually four C's that we can play, starting with middle C. This note has all the fingers down. In the next octave, almost all the fingers are up. In the next octave going up, you use the same fingering as previously. The top C, however, is really crazy.
C-sharp / D-flat
The middle C-sharp and top C-sharp are fingered the same way, with just the right little finger down. However, this note tends to sound very hollow because of the lack of fingers down. To counter this, I will add the right second, third, and fourth fingers to give the tone quality more body. This also helps stabilize the pitch of the C-sharp, which is notorious for going wildly sharp and generally out of control.
The D is one that is commonly mis-fingered. Note that the low D has the left index finger down, but the middle octave has that left index finger up. A trained flutist can tell if a flutist is playing the middle D with the left index finger down--it sounds stuffy and closed. Don't make this mistake! The top octave D still has the left index finger up, but the right fingers are up with the little finger down.
The E-flat is similar to the D. In the low octave, the left index finger is down, but in the middle octave, it is up. High register: The "all-fingers" note: Every finger that can press a key is down, including the left little finger on the "A-flat" key.
Flute fingerings for the low E and middle E are the same. For the high E, lift the left ring finger.
Low and middle Fs are the same. For the high F, lift the left third finger.
Low and middle F-sharps are the same. For the high F-sharp, lift the left third finger. Note that you cannot use the thumb B-flat and play the high F-sharp. The high F-sharp is often hard to play--it either doesn't speak or it screeches. Students often develop a fear of it. Just be confident and definitely don't back away from it. Keep a relaxed embouchure as well.
Low and middle Gs are the same. For the high G, simply take off the left thumb.
Low and middle A-flats are the same. For the high A-flat, lift the left thumb and left index finger.
Low and middle As are the same. For the high A, switch the index fingers: Left comes off and right goes down.
Low and middle register B-flats are the same. I find it quite delightful that there are three ways you can play this B-flat. Why was the flute designed this way? Who knows, but it's to your advantage to know all three and use them in the appropriate situation.
One and one - Left index and right index fingers, along with the left thumb and right little finger. Use this fingering when playing in sharp keys and situations where you don't have B-flats or A-sharps except as an occasional accidental.
Thumb - Slide the left thumb to the left key and you no longer need the right index finger. This is great when you are playing in flat key signature. Use it at all times in flat keys!
Left B-flat key - This is the one I have used the least. Some people swear by it, though, especially in chromatic scales. Try it and see how you like it.
The major recommendation for B-flat that I make is to use the thumb B-flat in all flat key situations. This avoids the use of the right index finger.
Low and middle Bs are the same. For the third octave, you get an odd fingering. Add left ring finger and play the two trill keys in the right hand. And you really must blow hard to get this note! Wimpy flutists need not bother.
If you have a B-foot joint, you can play the B below middle C. Keep a relaxed, very loose embouchure. Your right little finger needs to reach that last roller key.
Flute fingerings beyond the third octave...
There is a fourth octave ... sounds like the intro to Twilight Zone, doesn't it? Your flute fingerings chart might not even go that high. It is extremely difficult to play the notes in the fourth octave. It takes a lot of air support and they can sound like the most horrible shrieking noise you ever heard.
Flutists who use extended techniques often play in this fourth register and contemporary composers will write pieces with these notes--just because they can.
However, on a practical basis, you may never use these flute fingerings! So what good are they? In a learning environment, if you can belt out these notes, you have fantastic breath support. Other so-called "extended techniques" are good for practice purposes as well, including singing while playing. And this isn't nearly so horrible-sounding.