Delta used to have an option on their website that would show the cheapest and expensive dates on which to travel - so if your travel dates were flexible you could adjust for the cheapest airfare dates. But Delta no longer has this. I know some of the booking websites such as kayak allow you to search +/- three days. Do you know any websites that help you figure out best days for travel if you just know you want to go to a certain location at a certain time of year? Also, we are thinking of going into one airport and home from another, and don't care which is first, it would be nice to know if one option was cheaper...
Also, it would be nice if I could filter by airline.
Try Google Flights or www.skyscanner.net .We use Skyscanner for all of our travel through Europe and it has a great feature where you can select a date and destination "everywhere" and find the cheapest flights. You can also select a "whole month" option. Check it out!
Hopper.com. Tells you when to fly and buy for lower cost fares.
Huge thumbs up for both Google Flights and Skyscanner. Planned a recent 2.5-month trip to SE Asia, as well as an upcoming 2-month trip to Europe, and both tools were invaluable. If you have any sort of flexibility with your travel dates, the "month view" on Google Flights can save you a TON of money. Both sites will also potentially help you find options for O/W fares that are sometimes more cost effective. Good luck!
Actually pebblar (www.pebbar.com) now has the one-click flight time search feature and also one click access to google hotels (which is display together with Google flights as mentioned by Paul below). For your trips under pebblar, you can directly access those pages without having to search for individual cities.
I really like the features of google flight. A nice manual how to use it can be found here: http://www.huffingtonpost.com/2015/03/05/google-flights-ticks-better-travel-agent_n_6783692.html
I generally like www.momondo.com a lot. It also has the option of open jaw flights, where you fly in to one airport and out from another.
I also have a few tips on how to get cheaper ticket in general on my external blog, on this link http://travellingclaus.com/finding-cheap-plane-tickets/
My personal preferences are Kayak, Skyscanner and MakeMyTrip, in that order. I've been advised to check Google Flights, but haven't so far.
So, adding to the fabulous GFlights and Skyscanner answers below, I'm going to mention an even more esoteric way to find best dates for travel that also adds the best cities to travel to for when your destination is also Up in the Air :)
I am a travel free-bird, and will often answer my question of "Where should I go next month" by using this tool. It seems pretty powerful, but it's well hidden and almost like Kayak doesn't want everyone to know about it. I found it last year after tooling around.
How it works: You enter your starting point, chose the month or season (i.e. Sept 2015 or Fall 2015), and there are a few sliders to filter the results like max price and activity preference such as beach or ski destination, but it then responds with a map of the entire planet and shows you the best prices for every city that match your search.
I'm surprised nobody mentioned Expedia.
My numerous searches for cheap + suitable (!) flights
invariably end up with the best deal from Expedia.com
Google flights is my go to, they will show maps of places cost from your flying from and also give graphs per month based on how long your are planning on staying. It's extremely helpful when you know you want to go on a trip and that's about it.
The recommendations, as well as itasoftware, will cover published fares. But a travel agent can get you consolidator fares that are not available to the general public.