Train tickets in the UK are expensive. I recently flew from London to Glasgow because it was cheaper than taking the train, even counting the total cost including getting a bus from my home in Oxford to Heathrow airport, and a bus from Glasgow airport to the city centre. London to Edinburgh is about the same distance. There are plenty of flights from London to Edinburgh and you won't even have the expense of getting to the airport, since your plane from India will have already taken you there.
It is possible to get cheaper train tickets (called "Advance tickets") but they can only be used on the train that they're reserved on. If you miss your train, you'll have to buy a full-price ticket, so you'd need to leave lots of time to get from the airport to King's Cross (from where the Edinburgh trains depart).
But maybe you want to travel by train so you can see more of the country. There are some beautiful views on the section north of Newcastle and Durham Cathedral is another impressive sight. You can find information about the trains from National Rail and other sites. If you do use another site (maybe they have better user interfaces), use either National Rail or the actual train company's website to buy the ticket, so you avoid booking fees.
You aren't transiting in the UK, so you don't need a transit visa. Transit means entering a country just to make a travel connection to another country: for example, if you were entering the UK only to catch a flight to France. You're entering the UK to visit the UK, so you need a visitor visa.