In addition to working on tunes, you’ll want to spend some time on technique. One of the biggies is developing the embouchure, so here are some lip workouts for the ‘flute gym’:
Long tones: play your nicest middle D, then slur it to a C#, trying to make both notes the best you can. When you’re happy with the match, play the best C# you can. Then slur C# to C. Get a good C, slur to B, etc., on down to low D. You can also do this in the second octave, either progressing up from middle D, or down from the high B. (You can start this exercise on most any note. Low G is another good starting point.)
Octave slurs: Play a good low G, the let a high G grow out of it (if you want, fall back to the low G). Play a good low A, let the high A form, then do the same on B. Go back to the G, and work your way down to low D via F# and E. You can also combine the long tones and octave slurs into one exercise, if you like.
Then, to expand flexibility and range, play middle D, then C#, D then C natural, D then B, D then A, D then G, D then F#, D then E, D then low D. Then go up: middle D to E, D to F#, etc. The farther apart the notes are, the harder it is. You can start this exercise on any note and work up and down from that ‘home base’ note. For instance, start on low G, work down to low D, then work up from the G (G to A, G to B, G to C natural, G to C#, etc.) Harder yet, use low D as the ‘home’ note.
Harmonics: This is a little tricky at first, so don’t force it. If you can’t make it all the way to the top, don’t. Eventually you will be able to. Here’s the scale: D E F# G A B C# D, starting in the second octave (middle D) and playing up to the third octave D. Here’s how it’s fingered, low to high:
D xxxxxx
E xxxxxo
F# xxxxoo
G xxxooo
A xxxxxx
B xxxxxo
C# xxxxoo
D xxxooo
The only “real” fingerings you’re using are for the first four notes of the scale, then you start playing harmonics, instead of the actual note you’re fingering. Don’t blow harder for the high notes, just a bit faster (smaller lip opening) and aim the air a little bit higher (not as much down into the flute). This is a pretty good lip work-out, so make sure you balance it out with some lower notes.
Take your time, play as much of an exercise as sounds good, then stop. I wouldn’t use an electronic tuner all the time, but it’s good check your intonation from time to time, especially while you’re getting to know your new flute.
As for finger technique and style, my favorite tutors are ‘Fliuit’ by June McCormack and Conal O’Grada’s ‘An Fheadog Mhor’. June’s book is available in the US; you may have to order Conal’s directly.