Look at Generator hostel in London for shared room. I would normally say Z Hotels but outside budget.

Walks.com have many great tours that are reasonably priced.

I walk everywhere I can in London and you can look up free events on TimeOut London.

Not sure if I'd splurge on a show - I'd rather do the Jet Boat on Thames or the Harry Potter Studio tour (which is fun).

for food - EAT and Pret are good options as is M&S. Cafe deco near the British Museum (free) is cheap - I got a massive pasta dish and bottle of water for £5.50. I also created my own cupcake and chocolate walking tour in soho which if you have a sweet tooth is fun

For Paris, the street art walking tour is fabulous. You can buy sandwiches at a local Paul cafe.

Blablacar is a ride sharing service in UK and Europe to save money on transit

I have many London and Paris posts on my blog philatravelgirl as they are two of my favorite cities.