I had no idea there is a dry ski slope in Surrey. Depending on temps you should just need a jacket and if you have any sort of wind or athletic pant that should be fine. Definitely gloves to protect your hands when you fall. Their FAQs are pretty good.
Cool. You'll be fine. Instructors at dry slopes know what they are doing with beginners/restarters and the lack of speed with the surface stops catastrophic accidents. Just don't leave thumbs out or palms open when you fall as they have a tendency to get caught in holes in the matting. Gardening gloves are fine and jeans work well if the slope isn't too wet on the anti-abrasion front.
BTW I live maybe 10 miles from you tops if you are in Sutton. I learnt both to ski and snowboard on plastic. Snow is a doddle after you've mastered the Dendix, it's a lot slippier and more uniform.
No to notebook - you need to be concentrating on the lesson not cramming for a future exam - a lot of what lessons teach you is getting a feel for skis and an understanding of how what you do affects the way the ski makes you move. It isn't prescriptive rote learning.
Don't worry about rentals - the dry slope will sort all that out or the nearest shop when you do travel to the alps.
But as others have said the most important thing is enjoy - make sure you laugh at all your failures and celebrate your little triumphs. I sometimes envy beginners - they can get the most fun out of a day when po-faced "experts" are throwing tantrums about how conditions sucked.