Go to national rail enquiries.
You can get an off-peak ticket for £16 return for travel on that Saturday. I don’t know if it will go up on the day. Regarding the tube, you can buy a ticket including an all-day travelcard for about £22. If you don’t have an Oyster card and will make more than 2 journeys by tube, this is the cheapest option.
You can sometimes get “advance” single tickets that are cheaper but you must get the train service specified on the ticket. These are cheapest over 2 weeks in advance (and sold up to 3 days before travel). But since you are travelling off-peak I suspect you won’t get cheaper than the off-peak return.
If you are with any cashback site, you can book rail tickets through any rail operator and get cashback with quite a few. So you can save a few percent as long as you are willing to pick up the tickets at the station using the credit card you booked with online.