Hello - I have a little bit of experience with Royal Caribbean although my kids were a bit younger.
You don’t have to spend any extra money but you will probably want to.
The ship’s shore excursions are the easiest option but not the cheapest. You can arrange something independently with a bit of research. Have a look at the port websites or port forums on cruise critic. Some ports you can just wander off the ship and be in the town (Barcelona) and others are miles away (Rome).
There is plenty of included food but you pay extra for a specialty restaurant which is a change and maybe a bit nicer. Free drinks include tea, coffee, squash and water. If you want to drink a lot of alcohol then look out for the offers on drinks packages you can get in advance.
There is a main dining room where you will be assigned a table for dinner on an early or late sitting or you can be on my time dining when you book a time each night. You will get to know your waiter and can eat as much as you want from the menu. Breakfast and lunch is usually in the buffet. I think there is another included cafe too which is quieter especially on embarkation day as not everyone knows about it. On sea days there can also be breakfast and lunch available in the main dining room.
You can also spend money on photos, shopping, spa treatments and laundry (on some days they have a cheaper wash and fold service - we were stung paying by the item a day or two before)
I was worried it would feel claustrophobic and busy and like we were being herded but it didn’t feel like that. The pool area can be quite busy but there is always a quiet bit of deck you can find or maybe get your own balcony if private outside space is important to you.
From the UK I think you would be on Anthem of the Seas so I think the balconies forward of the lifeboats would have the best views down to the sea and look out for spacious ocean view rooms in the front corners if you are thinking of sharing although you may want two bathrooms with teens.
lots of varied entertainment available. Shows, music, karaoke etc. and activities like Rock climbing, skating, laser tag, flow rider - that is all included. Mine loved the kids club but as I say they were younger. The advice I have seen for teens is to make sure they go to the club on the first night to meet people.
cons - you don’t stay long anywhere so it is just a taster of each place. It is all a bit fake looking. Not sure what I expect it to look like! Everything starts exactly on time and there was lots I wanted to do so I did get a bit stressed rushing to activities but nothing is compulsory.
Generally though we loved it and the kids loved it too. There was something for everyone. I did a short one when the kids were older and left DH at home - we stuck together that time so didn’t try the teen club and we all enjoyed it.