Hi Karin
I did this trip this time last year - you will have a fabulous time.
No one in Iceland dresses up, it's all really casual - dressing for warmth and comfort is the priority. If you are taken to the same restaurants that we went to (and I hope you are because the food is out of this world), one is quite rustic and the lightening very low, whereas the other is very bright and airy and feels like an elegant Georgian dining room, but again, everyone is in jeans and sweaters.
As Monty says, it can be very icy and slippy underfoot, so I'd suggest comfy walking boots with a pair of detachable grippers (Yaktrax or similar) for when you as out of the town. The Golden Circle (unmissable) will probably still be very snowy and icy. It is also incredibly windy, so I'd recommend hat and scarf or a balaclava and some thick gloves. You'll be on and off the coach quite a bit, so wearing layers is the way to go.
Have a super time - happy to help if you have any other questions
All the best
Mandy