How fun that your daughter is showing an interest in music at this age, and how supportive for you to be looking at what to do!
I have been playing the trombone and ringing English handbells for more than 25 years each, have a B.A. in trombone, am self-taught on the piano, and worked for 8 years in a music retail store which rented primarily band and orchestra instruments. The question you ask is one I have had posed to me hundreds of times by parents, so I'd like to offer some feedback.
The most important thing for a child interested in music at this age is that it is FUN! Believe me, I know all about practicing and lessons...but SO many kids get turned off from music at an early age because of the pressure parents put on them.
I've seen it over and over, in teaching students of my own as well as watching those that used the services of the company's music studio. I understand that it's important to set boundaries and encourage children to finish whatever they start; however, at this age, if a child loses interest and is made to continue, the results are often disappointing.
At the very least, it is a waste of the teacher's time and the parents' money if a child doesn't want to play, has to be forced to practice and doesn't like/want lessons. Sometimes, it carries even further than that, and the child is completely turned off from music altogether.
True, there are benefits in terms of brain development, discipline and mastery that come with music...but the expectations should always be measured by the age of the child. And, statistically, at such a young age, I would expect a child to move around and explore different options. That's just in their nature at this point.
If you had come to me at the music store, I would most definitely have discouraged you from starting your daughter on a band instrument (flute, clarinet, trumpet, etc.) for a couple of reasons...
First of all, she would almost certainly be too small physically to be able to do what is required to play any of those instruments. They aren't sized the way stringed instruments are, and it's usually 5th or 6th grade before kids are big enough to play them (and my arm never felt like it was long enough for the trombone slide, even years later!).
The other reason is because band instruments are usually played in a group setting. Statistics we saw from different studies on this issue actually showed that students who started a band instrument a year or two (or more) ahead of the "norm" actually had a higher drop-out rate than those who started with the rest of the group.
The reasons considered had to do primarily with the fact that it's not very rewarding for most students to practice and practice and practice -- and then not have a concert to play in, etc. Also, they tended to get bored in band class b/c they had already learned how to play the instrument, so they weren't challenged enough to stick with it.
Piano and the smaller-sized stringed instruments are okay choices at this age, if that is what you're interested in. Maybe it sounds silly to find a teacher first, but I'd actually suggest you start shopping for either/both and ask them questions about how young they will teach and what they will expect.
I've known piano teachers who start children as young as 5; others require them to be a little older, or to attain a certain level of reading. Some are okay if you have a keyboard at home to practice (let's fact it -- they're not going to be playing the Rachmaninoff Concerto No. 2 on the first lesson!) while others will want you to have the full 88 keys available.
Many parents like the smaller-sized string instruments (once the children get past that "squeaky" phase) and a rental program is definitely the way to go for one of those because they usually offer exchanges to the next size with little or no cost to you. That certainly beats shelling out $500 or so for each instrument as you go up -- and the resale profit on most used instruments is minimal.
Some parents like the Suzuki method for strings b/c it gets the child playing right away. Some teachers I know offer criticism on the instant gratification...and also often find it challenging to help those students transition into reading music later. Pros and cons on both sides, obviously.
I don't know if you've checked out local music and movement classes through KinderMusic or Gymboree Play & Music -- or whatever others are out there. That might also be a way to whet her interest and give her some options before plunking down serious money.
There are also "kid friendly" instruments (read: cheap) in many toy catalogs. I've seen guitars, recorders, drum sets, hand-held percussion, harmonicas and the like. Personally, my son LOVES watching me play bells and always wants to get involved when I practice at home. So, for his birthday this fall, I shelled out a whole $30 and bought him the brightly-colored, metal, tuned set of 8 bells and he can play along, too! He loves them, and they are quite durable. They've already been dropped a number of times and are still going strong.
Best wishes as you pursue options that will work for your child, your budget and your family. I hope that you find a creative solution that is fun for her and affordable for you. Feel free to send me a message if you'd like to discuss this further.