Travel by Train to Scotland and Wales

I'm going to St Andrews in April. There is no train to St Andrews, you have to go by bus or car.
I found an easy way to find routes between cities by using Google maps an get the directions between the two stops. It should list your options if train or bus routes are available.

Trains.....there is really no such things as Britrail anymore.

National rail is the website we suggest people use to find out the train schedules and what trian companies operate those routes:

http://www.nationalrail.co.uk

Scotrail is a train company that operates trains in Scotland.

As an example, for your route from Cardiff to London. When looking at the Nationalrail website you would see that trains leave from Cardiff Central Station and go to London Paddington Station. They are operated by Great Western Railway.

The first thing I suggest you do is go to the Man in Seat 61 website. This is the definitive guide to train travel. The site is heavy with information on UK train travel.

When you say Britrail I assume that you mean the railpass called that rather than a train company.
The short answer is, almost certainly not. With Advance ticket prices being what they are now point to point tickets will almost certainly be cheaper.
Even your very long journey from Edinburgh to Cardiff should be around £50 each if purchased as Edinburgh to Crewe and Crewe to Cardiff.
Buy 2 x Two Together Railcards for £30 each and you will save 1/3 off all travel after 0930.
You can buy a rail ticket to St Andrews which includes the fare on the short and frequent bus link from Leuchars station. EDIT- The Station Code for St Andrew's is SAO

EDI to SAO Day Return on any train £26.10, buy on the day, advance fares restricted to timed trains are only £4 cheaper
EDI to INV Period Return £38.40 Advance Fares
EDI to GLA Day Return (on any train, peak or off peak) £20, buy on the day
EDI to Crewe on the 0852 £27.60 Advance Fare
Crewe to Cardiff on Transport for Wales £30.90 on the 1310, or £39.70 in 1st class (this is proper 1st class and well worth the modest upgrade. For a modest extra charge you can have lunch service, one of the best and most modestly priced meal services on a train anywhere in Western Europe) Advance Fare, It is worth the 1 hour wait at Crewe to catch what is colloquially and semi officially known as The Gerald Train, rather than a straight connection into the 1210 train- even in standard class.
Cardiff to London Paddington £36.50 Advance Fare after 1018

TOTAL £180 before Railcard reductions. The railcard even works for 1st class fares.
Add back the £60 for the railcards and it is about £135 per person in rail fares.

On Cardiff to London Paddington you can book now, on the other sectors where Advance Fares are available (Inverness and EDI to Cardiff) you can book from 8 weeks out, but don't stress about exactly 8 weeks out- anytime between 8 weeks and a few days beforehand will do.

Just book all the advance tickets with LNER (even though you are not travelling on them)- much the most friendly interface for American guests.

There is no train to St Andrews, you have to go by bus or car.

Not quite true, at least in practice. One can take the train to Leuchars and get a taxi into St. Andrews. It's no more than a 10-15 minute ride.

Thank you all!

One can take the train to Leuchars and get a taxi into St. Andrews.
It's no more than a 10-15 minute ride.

You can also take the no. 99 bus, and the ride is about the same duration.