"cheap' is not the word you can use for this trip. You are literally going to the most popular places at the most busy time, during a fuel crisis that is inflating airfares and potentially train prices soon.

That being said, you could have an AFFORDABLE Europe trip but you need to set your expectations fro budget.

FLights will probably be $1200+ round trip LA to London (which is USUALLY the cheapest gateway city to Europe - but not always).

Eurail pass will probably cost you +-$450 depending on how many travel days you actually need. THen for countries like Spain you will need seat reservations, and in other countries if you use the high speed ICE/CE trains. Seat reservations will cost an extra $5-30 per reservation when needed. To avoid these prepurchase your train tickets for each destination (ie point to point tickets) or focus on REGIONAL slower trains instead of High Speed ICE/CE trains. Take a bit more time and connections, but will save you money on reservations. Point to Point tickets might also be cheaper than a Eurail pass if bought early (ie now) but will limit your ability to be spontaneous since you will have to know your route and buy all tickets ahead of your trip for the best price.

Hostels will vary from an average of $30 per night to $90-100 in some popular cities at the most popular hostels.

Food can vary wildly depending on where you eat, how lax you are about sticking to a budget, and how much alchol you drink. While you could get by with about $45 per day eating if you have a small cheap breakfast (ie coffee and croissant), a picnic lunch from a market or grocers or train station (ie a baguete with lunch meat, cheese, fruit from a grocery store) and a small cafe or local restaurent dinner every couple days, and zero alcohol, to $75+.

Sightseeing will be where you won't want to skimp much because it's legit why you are probably going to Europe in the first place. Forget unique experiences if you want a 'cheap' trip. Find student/youth discounts (if you are under 27 adn/or a student) plan to be in cities when your must dos offer free days (paris's louvre offers this still i think one day week), look in to city museum pass a(again paris has one that could save you money on sightseeing), accept you don't need to go inside every attraction and prioritize which ones you MUST see/do on you budget. take free walking tours to see a city. THis could easily be an average of $25-50 dollars per day depending on what you expect to see or do.

Walk. In cities walk or take public transportation instead of more expensive taxis and ride shares. The costs for taxis and ride share will add up quick. And walking / public transportation is super convenient in European cities.

Consider mixing in some TOWNS into your itinerary. Not only will they be cheaper than cities and help keep costs down, Europe's towns in every country are amazing and provide a more intimate look at the country's culture, are less crowded (mostly), and will carbonate your trip with a different view of Europe.