Hampton Court Palace?
 

Looks like trains run from WAT ever 30 minutes or so. Should these be purchased in advance? I would expect we would be on the train at 0857. Thank you

Is the wait long for tickets. How early should we arrive?

there are plenty of ticket machines.

they take notes as well as coins.

to make sure you get the 08.57, i would aim to be there for about 08.40. this will give you time to get the tickets, find the train, and get seats, even to buy a cup of tea and bun before you get on.

There's no need to buy in advance for a local commuter service like this.

Absolutely no neeed to buy tickets in advance for such a short journey - just turn up and buy tickets on the day.