Some great advice above. Especially from Carol:
I think your biggest challenge is to start with what you want to see
and where you want to go. Once you know that, people on the forum can
easily help you figure out your transportation options.
As Jennifer said:
For your accommodation, you will only find hostels in larger places,
other than the YHA, which is rurally based and designed for walkers.
In towns, the Premier Inn chain is as good value as you will find,
although due to its dynamic pricing, many places will already have
sold the cheapest rooms for September/October, particularly at the
weekend. What is your nightly budget in GBP?
That last question is important. My favorite tool for converting dollars to GBP is https://www.xe.com. (Remember, that's the optimum exchange rate and consumers get a slightly less favorable rate; that's just how it works.)
Also good to know, as Jennifer pointed out, many of the cheapest rooms are already sold out for September and October. For example, I stayed at the Bar Convent in York a couple of years ago. It's one of the best values for money I've ever experienced in lodging (breakfast is included!). I checked their calendar for the first week in October. Their rooms are mostly already booked, especially the cheapest single (£40). https://www.bar-convent.org.uk/accommodation.htm
I'm more of a city girl, but was based in York for a few days and enjoyed walking around to all the interesting sites there, and a few day trips (by train).
If you don't mind chilly-to-cold weather, you may find more accommodations are available in November. But a cold rainy day doesn't seem to me to be ideal for enjoying the countryside.
As to transportation, besides the rail system, there are inter-city buses (in England they are called "coaches" while "buses" generally mean in-town buses) which if you're not in a hurry can give you some interesting rides, with some attractively low fares. The website for the national system is here: https://www.nationalexpress.com/en. Maybe our UK friends can chime in with info on any local bus/coach systems once you develop your destination wish lists.
I love visiting the UK, and encourage you to work up an itinerary for your trip, and make it happen. There are lots of tips for money-saving travel on this forum and elsewhere online. If I'd waited to travel till I got rich I'd have never gone anywhere!
P.S.: I may be stating the obvious and if so I apologize, but don't forget that you can check out travel guides from your public library, a thrifty way to narrow down your wish list.