England rail travel

You will get a significant discount if you book in advance but that will tie you to a specific train in each direction. Tickets that can be used on any train are much more expensive. The site for the specific company is http://www.virgintrains.co.uk/ but if you have any problems other people here say that http://www.eastcoast.co.uk/ reliably accepts US cards. The other problem you have is that there is sometimes rail engineering work on Sundays. There should be details on the Virgin site under the Travel Updates link at the top of the page, but often they aren't announced until the beginning of the month.

Is it necessary to buy a ticket ahead of time?
Only if you want to save money by making advance purchase of a specific train on a specific day - unchangeable. Do trains ever get sold out? No, but particularly on football days (sometimes Sunday) and at rush hour (not Sunday) they can get very overcrowded. Many a time I have had to walk over people sitting on the floor between cars. Last Wednesday was the last time. If we buy an "Offpeak" ticket early, can it be used on any "offpeak" time train? Yes and no. Tickets are issued for a particular date and route. Yes, you can take any off-peak train on the date on the ticket on the specific routing on the ticket. No, you cannot travel on a single or outwards on a return other than the date on the ticket and you may not break your journey en-route. If it is an off-peak return you have 30 days to make the return journey and you can break the return journey as long as you travel in a straight line and don't backtrack. If you walk up and ask for a ticket you will get one with the current date on it UNLESS you request a specific date. BTW - It is the "Lake District", not "the Lakes".