I think they'd probably consider plimsolls to be trainers, they don't need to be really smart, they just need to not look like sports shoes, I normally wear Timberlands.
Ah I see... so how do you get into these places? Are they only open on matchdays or other days too? And do you still watch the game from your normal seat?
A Premier Club membership gives you access to the members bars/restaurant, seats just outside of those bars in the West Stand Lower and a parking space right next to the ground. You pay a club membership annually(of a single life membership), then you pay for a five year season ticket.