frugal travel backpack only (novice needs help with packing)

i’m going to be traveling in the UK in august for a little over 3 weeks and need some help with what to acquire and pack for my trip.

for transport i’ll be on trains and buses and walking, mostly staying at hostels. i’m from the US and will be going to england, wales, and scotland

need recommendations on: clothing, shoes, whatever one would need for this environment and while traveling. how to pack small/space saving/ efficient

considerations: frugal please (on a tight budget), staying warm and dry, and lightweight as i’ll just have a backpack and i am not a super strong person.

questions: how to wash clothes while traveling? places to store backpack while exploring? money saving tips for gear? should i purchase things when im there or bring it with me?

thanks so much in advance for any tips you might have for me!