First trip with plane EVER and to United Kingdom - dream come true!

So I've written one post since I got here, and other two were just my illustrations - then I took a break because I went home and well, I was so excited about my trip this weekend that I could not think about anything else.

BUT, I forgot about the point of this platform - and that is to write.

I've been in love with United Kingdom since I was 12 and discovered british music. The Smiths took my breath away (still do - and are currently on my phone so I can listen to the Morrisseys voice when driving through UK. I'm travelling with my ex-roommate who is currently living in the UK (talk about meeting the right people). The plan is to visit London, Edinburgh and Glasgow in 10 days, and also stay in the city where she lives for a couple of days.

(GUYYYYYYYYYYYYS, I'M SO EXCITED, MY THOUGHTS ARE RUNNING A MARATHON!!!)

I've been planning what to bring since like mid of February, while I'm going with just a cabin baggage which makes things a bit hard - but well - minimalism I guess.

Since I can't really bring the words out on how excited I am - I'll fire up some questions and maybe some experienced travellers can suggest some cool places which I might miss. (I really don't want this to happen).

1. What places are not so touristy in London?
2. Which touristy places are really WORTH visiting?
3. What you can not miss in Edinburgh?
4. I'm visiting Glasgow just for a day - what is the most important thing to see?

P.S. I'm going to try to write each day SOMETHING about the trip/or at least post some pictures.
P.P.S. Sorry for boring edit and possible spelling mistakes, I'm writing this on my job (shhhh) on which I can not focus because I'm so EXCITED.