GA is very safe these days. I give somewhere between 3 and 15 a day (depending on whether the list is lots of short cases or a few longer ones).
Depending on which hospital you're at, you will either be given your anaesthetic in a small room just next to theatre and then wheeled through, or taken straight into theatre and anaesthetised in there (all our patients are anaesthetised in theatre but some hospital when I was training still used anaesthetic rooms).
You'll be seen by the anaesthetist preop, who will ask about medical conditions or current prescribed medication. They will assess your airway (mouth opening and jaw movement). Here we ask patients to sign consent for anaesthesia, in the uk 5 years ago we didn't but that may have changed.
You'll have monitoring attached to look at heart rate and rhythm, oxygen levels and blood pressure. A small cannula goes into a vein (I use local anaesthetics through a tiny needle first to numb the skin before I put the cannula in) and the staff go through a checklist confirming patient identity, procedure and site, allergies (WHO checklist) before you're given drugs through the iv line which have you asleep in 10-30seconds.
I don't use gas inductions on adults unless there are very specific (rare) airway issues. You may be asked to breathe 100% oxygen through a mask before you go to sleep but it's not compulsory and if I have patients who are terrified if things over their faces (with otherwise straightforward airways) I don't put the mask on until they're asleep.
For very anxious patients I still use benzo premeds (lorazepam or temazepam) which can help. Mostly I talk to them, reassure them and they're fine.
I'd be happy to explain what happens next if you would like to know but equally happy to stop there!
Once surgery is over they turn off the anaesthetic and you wake up. You're wheeled to the recovery room where a nurse monitors your conscious level, vital signs, gives pain / sickness meds if needed. From there you go to the ward or daystay unit depending on whether you're going home that day or staying overnight.
YANBU to be scared, most people are. It's very very normal!