My DD11 (almost 12) started taking lessons about 2.5 years ago. She has both an electric and an acoustic.
I think you have to get both components correct for your DD to have a good experience.
First, the guitar. We bought the acoustic from the Guitar Center, which I believe is a chain, with the advice of the guitar teacher. There are lots of mom and pop stores to purchase from, but we wanted to go somewhere with a big selection of both used and new. This worked for us.
She ended up with a Yamaha that has a built in tuner and it is a 7/8 scale (I think that is what it is called). It is slightly smaller and thinner than a full-sized guitar but much larger than a "child's" guitar. I think it cost around $200 new. The people at the Guitar Center were very helpful. They didn't try to upsell me, and they gave me the pros and cons of every guitar she liked. They played every guitar we were interested in so we could hear the sound difference. And her teacher thinks it was a great choice. Another brand they told me that is good to learn on is an Ibanez. I think Daisy Rock, mentioned by a pp, is the brand Miley Cyrus plays. My daughter has her eye on one, but she can't get it until she practices more (more on that later).
Another pp mentioned the Line 6 brand of amps for the electric. We have one too. It also costs about $100, but it sounds great. This is the brand DD's instructor has in his studio--though he has a much bigger, better, pricier one! Again, the Guitar Center had a lot of new and used to choose from and they plugged a guitar into every one that we were interested in and played so we could hear it. To be honest I forget the name of the other brand that we almost chose, but it was in the same price range.
The teacher is the 2nd component. You need to make sure it is a good fit, and he or she will teach her the things she wants to play. DD's instructor has played since he was 11 or 12 and has a degree in music from Berkley in Mass. which is a pretty prestigious music school. He has toured with bands and currently still plays in a band on the weekends. He's in his mid 40's and is an awesome guitarist. I sit in on the lessons just so I can hear him play. He has a great ear, and when my daughter tells him she wants to play a song he's never heard of, he looks it up on
youtube, listens to it once or twice, then writes down the chords for her and they are off to the races.
As far as the actual lessons go, my daughter's teacher is really cool in that he teaches from the Mel Bay books (which have been standard guitar lesson books for decades), but he also teaches songs at the same time, so it isn't boring. Right from the beginning he taught her Green Day--now Good Riddance is a staple in her repertoire. She also plays Taylor Swift, Miley Cyrus and Colby Callait.
Her teacher believes that it is important to learn to read music and believes it makes for a better musician. She likes the chords and songs better, for sure, but she knows her scales and can sight read.
Our only frustration is that she doesn't practice enough, but loves it and doesn't want to give it up. She would be just awesome if she practiced more.
Good luck with whatever you decide.