There are no trains to St Andrews. If you want a multi-modal journey, the usual route would be train from Edinburgh to Leuchars (just over an hour) and then bus to St Andrews (about 15 minutes). It is not necessary to reserve seats on these trains.

Alternatively, you can take bus X59 the entire way from Edinburgh to St Andrews. This is served hourly, usually by large comfortable coaches but not always. It takes 2 hours and costs £8.10 for a return ticket. You can pay the driver when you board, contactless cards accepted (or download an app and buy there if you really want to).

Meanwhile, the fare for a train+bus same day return via Leuchars is £32.80. With this ticket you can take any train you like except before 9.15am on working days. It does not sell out and the price does not change, so you can buy it just before you start the journey.

It's possible to get the price down by booking the train and bus separately, but in this case you do have to pre-book as the cheaper rail fares can sell out. One-way tickets between Edinburgh and Leuchars seems to be available for most of next week on Scotrail for £10.20, roughly two departures an hour, but this is a fixed time ticket so you are required to take the exact train that you booked. LNER has an outbound ticket for £8.30 and inbound for £7.30, but for a day trip there are only two options of train times per day.

The individual bus fare for the short journey from Leuchars to St Andrews is £5.10 in each direction, which can be reduced to £9 return by asking for a "North East Fife Day Rider", but is still more expensive than taking the bus all the way from Edinburgh.