Most people choose by sound, which is a good reason. These are the practical differences nobody mentions until month three.
Saxophone
The quickest to a satisfying noise. Fingering is straightforward, the first octave comes fast, and a beginner can play a recognizable tune early, which matters enormously for whether they continue. It's heavy on a small frame, it's loud in a house, and reeds are a running cost. Alto is the standard starting size, and tenor is a lot of instrument for a young beginner.
Clarinet
Harder for the first two months and better for the ten years afterwards. The break between registers is a genuine hurdle that saxophone doesn't have, and getting past it teaches control that transfers to everything. Light, quiet enough to practice in an apartment, and the standard route into concert band. Also a reed instrument, so the same running cost.
Flute
No reeds, which is a real saving and one less thing to go wrong. The difficulty is entirely at the start: producing any sound at all takes most people longer than on either reed instrument, and some never quite click with it. Light, portable, and the quietest of the three to live with.
Switching later
It's normal and it's not starting again. Fingering transfers substantially between saxophone and clarinet, and reading, rhythm and everything you know about music transfers completely. Plenty of players end up doubling on all three. Choosing "wrong" at ten costs a few months at most.
Buying
A school rental or a decent used student instrument is fine to start on. What's not fine is a very cheap new instrument from a marketplace listing, which frequently can't be made to play in tune by anybody, and which convinces a beginner they have no ear. If you're about to buy something and are unsure, send me the listing. That's ten minutes of my time and it saves the wrong purchase.
Once you have one, what happens in a first lesson covers the rest, and lessons run in person around Cincinnati, Covington and Newport, or online.