I have most of my information (like emails of correspondence with lodgings) on my Moto X these days, but I do travel with some paper. I organize by day for the most part, although I may have some categories that are each organized that way. It depends on the trip and situation.
I used to use plastic sleeves, folders or small accordion files, but I've gotten the paper down to maybe a half-inch. I do toss as I go, making sure to flush (if possible) the critical data torn into tiny little bits of paper. However, I also collect as I go -- things like the notes others have mentioned, business cards from restaurants, and especially receipts from credit card purchases or ATM transactions. I do like to keep track of what we spend, so that I can add up what the total trip cost was at the end.
To keep things somewhat organized without a bunch of separate envelopes running around my carry-on, I use an Eagle Creek 041152 packing cube like this one in the strobe green. The cube itself is plenty big for all my paper on the way over and on the way back. It is practically weightless and the color makes it easy to find in my luggage and hard to leave behind in the room.
Part of my preparation for the trip is to make a spreadsheet with each day that includes the following columns: Night (as in night #1 of the trip), Day (as in day of the week), Date, Where (as in the city, usually indicated by airport code, but not always), What (general information on hotels, travel details), Lodging EUR, Lodging USD, Transportation EUR, Transportation USD, See/Do EUR, See/Do USD (all those are mostly for planning purposes, although the columns with the right formulas can be used to put in the actual expenditures when I get home), See/Do What Notes (details of ticket numbers, hotel reservations, what's been pre-paid, what I'll need to pay in cash, what there is to see or do, how long that is likely to take and how much it is likely to cost -- you get the idea). In essence, my spreadsheet is a summary of the whole trip. It's not always perfect and subject to change on the fly, but it is a good guide for me.
For security reasons, bulk reasons and because I read that we should not copy all our cards, front and back, I do a spreadsheet that includes all our important numbers. That actually cuts all the info down to one page and includes columns like Name, Number, Issue Date, Expiration Date, Business Hours USA (phone numbers), Urgent 24/7 (phone numbers). The rows include things like our passports, credit and debit cards we take, AZ driver licenses, IDPs, medicare supplement cards, etc. This is only for the critical cards we actually take with us.
And I do a table that I send to our kids, some friends and put in an obvious location in our carry-ons. It includes emergency information like our names, home address, mobile phone numbers and the following columns: Dates (when we go from one location to another), Enroute (the names of those from-to locations, usually but not always using airport symbols) and Where we will be (with detailed information on the lodgings including name, phone number, contact person, address, and a link to the place).
I realize this is a lot of upfront planning, but I like to be able to get as much as possible on 1-2 sheets of paper, then use hard copies to show for check-in at the lodgings, if needed. Now that I have the spreadsheets and tables set up, I can copy and paste and change them for each trip.