British trains are rather expensive, however there are some options.
If you're a European (but not U.K.) citizen you can buy a 3, 4, 6 or 8 days within 1 month Interrail pass. This allows unlimited travel across on any train for the respective number of days within the UK (A 'One Country' pass). These are cheaper if you're 25 or under (from EUR144 to EUR221) as opposed to 26 or older (EUR212 to EUR329).
For some reason the US / rest of the world version of this, the EURail pass, doesn't include the U.K. Instead, EURail pass holders get a 50% discount on BritRail passes. There are two main types of BritRail pass (travel 3, 4, 8 or 15 days in 2 months, or travel 3, 4, 8, 15, 22 days or 1 month consecutively), with added discounts for being 25 or under. Not only this, but there are regional versions, including:
- Scotland, England, Wales
- Scotland Central Belt (Glasgow - Edinburgh)
- British Isles (U.K. + Ireland)
- South West England
Assuming you get the 'Scotland, England, Wales' pass, the prices I looked up varied from $109 (EURail pass holder, under 26, 3 days consecutive travel) to $675 (non-EURail pass holder, under 26, 1 month consecutive).
If you're going to Paris you also get a Eurostar discount with the BritRail pass.
Alternatively if you want to keep it structured you can get some great prices on from MyTrainTicket and The TrainLine. Tickets are cheaper the further you book ahead, and also when you avoid 'peak' time (usually before 9:15am).
Finally, never buy tickets at the station. You will almost always pay an absolutely exorbitant amount.