Aww, sounds lovely, I'm sure you'll have a good time. I would have thought at a golf club in the afternoon it will be 'smart casual' so probably something like a nice summer dress perhaps with a jacket/cardy to cover your shoulders and smart sandals would be fine. I wouldn't have thought you need to particularly glam up unless you want to, probably just no shorts/flip flops/torn denim (why is torn denim still on dress codes, has anyone worn torn denim since the 90s!). Def try and find out if you will be sitting inside or out as you don't want to be freezing!
What happens, it might slightly vary from place to place but every tea I've been to, you are greeted at the dining room entrance and seated by a waiter as per a 'normal' restaurant, the seating might be at regular tables or more informal sofas/armchairs and low tables or outside more 'picnic' style. You'll be asked what hot drink you want (usually coffee, hot chocolate, herbal teas, sometimes other soft drinks are alternatives for non tea drinkers), and will be brought your drinks first along with your bubbles. If it's a particularly naice place you might be offered a 'tea menu' of lots of different kinds of loose leaf tea, I stick with english breakfast/builders because I'm common lol 
Depending on the poshness of the place it might be served as tea bags in a pot or loose leaf with a strainer, but I'd expect nice china either way...
The food will usually come on cake stands to be shared between your whole party, sandwiches, savoury pastries, mini quiches etc as savoury and little cakes, sweet tarts, scones with jam and cream etc for sweet. Usually you get a little menu saying what's what, and explaining any dietary stuff. Sometimes savoury comes first then sweet, sometimes all together. Sometimes you get 'refills', sometimes just one round. Just ask if it's not clear! Usually you can sit and take your time over it, sometimes there are free newspapers or a musician playing etc. so don't feel pressured to rush through.
You should be able to ask for a doggy bag of anything uneaten. I usually eat it all because I'm greedy and 'mini' things slip down easier than full portions, despite MN's general assertion that afternoon tea is hugely extravagant and contains 'lashings' of everything I don't find it a huge meal compared to going out in the evening and eating 3 courses... but obviously depends on your appetite and how generous the place is. Maybe have a light lunch or skip altogether in anticipation?
Enjoy!