I think with a lot of travel agents, you have to be, something to do with under 18s not being able to enter into a legal contract. You should able to book with parental permisson though. You could book flights without going through a travel agent, you need to check which ones allow you to fly (I think Ryanair do),and book acommodation yourself, again would have to check if they will accept you.
Booking into a hostel would be cheeper, you can do it through hostelworld.com. Of course if you do it without going through travel agents, then I'm not sure where you would stand with travel insurance. My friends and I are looking into doing this for a skiing trip in the winter, so I have to find everything out too. My cousin did, and was all good, I'd ask her exactly what she done, but she's in Japan at the moment lol
You are absolutely right. Need to be of age. If you are looking for a country in which you can go quite young and at the same time try everything, I advise Russia. Moreover, there are many beautiful places and landscapes. Pay attention to this site https://russiavoyager.com/. This company is engaged in accompanying tourists.