Flute and trumpet lessons

[Hopemore]

Hi.

My 6 years old has been asking for trumpet lessons. I think this is because they do it at school at some point (not sure if it is in Y3 or 4).
Apparently she can't wait.
She also likes flute.

So I just want opinions if she is in the right age, if it is worth paying for private lesson for her to try out and where I can buy a flute for her to play at home (not a toy one). I have seen few ones on eBay, from Germany and China, but I am not sure if I should buy it or not.

TIA

Please bare in mind I am totally ignorant at these instruments

So the fife arrived today and DD is so disappointed that she can't get a sound out of it and doesn't know how to place her fingers. Also she says she doesn't want to blow on the side but at the end of the instrument iykwim.
I have tried some youtube but I think she is just tired right now.
Will search for a lesson or two to give her a start.

How about a Clarineo? We have just got one for DD as she is too small for a clarinet. I would recommend going to a woodwind specialist though - the place we purchased from a staff member showed her how to hold it, how to use her mouth etc and really encouraged her. She left the shop grinning from ear to ear knowing that she could make it work. If we had bought on line that may not have been the case.