I love Momondo! I use it for all of my flight research. They have the best search options and filtering tools out of any flight site that I have been on:
You can search for one-way flights, return flights, or even perform multi-city searches (I use the multi-city function if I want an extended layover somewhere, or if I'll be flying into one city or country but out of another).
Plus you can filter by arrival and/or departure times, maximum flight duration, and maximum number of stops. Also, they are one of the few flight search sites that includes most of the budget airlines! Makes searching for flights easy peasy!
Expedia is another great research tool!
You can search for flights, hotels, vehicle rentals, all-inclusive vacations, and even cruises!
They also have a 'Things to do' tab which is really awesome for trip planning because it shows you a bunch of tours and excursions that you can partake in whichever city you wind up!
Plus you can filter by time slots (early morning, morning, afternoon, or evening), number of stops, and you can sort by price, duration, departure times or arrival times.
I love SeatGuru - it's a website that has airplane maps for most of the larger airlines (and some of the smaller ones) - and it tells you the desirability of the seat location. This is especially beneficial for long-haul flights when you want to know where the best seat on the plane is so you are not stuck sitting next to the bathrooms for 13 hours!!!
In order to keep flight costs down, I search for low-cost budget airlines that fly to the particular destination I'm headed to. For example, if I'm off to Lisbon, Portugal - I will begin my search with flights to Lisbon, but then I will search for cheap flights to Europe and then check the budget airlines to find the cheapest flight to Lisbon. On my last trip I flew through London because it cost me $51 GBP on Monarch to fly from London to Lisbon and the flight to London was much cheaper than trying to book straight through to Lisbon, and then I found a cheap flight home with Azores Airlines. Keep in mind that this likely also will increase your travel time to the destination. ***With a caveat - I usually overnight in the layover destination JUST IN CASE the first flight is delayed, because if you do not book a connecting flight with a partner airline they are not required to compensate you if you miss your connection. See the section on Missed Flight Connections. I factor the cost of the hotel in or whether there is a friend I can stay with in the layover destination when trying to determine if the cheaper flight will be worth it. Also keep in mind that the budget airlines also tend to have much stricter baggage policies and charge ridiculous fees for overweight bags or for bags in general and in some cases for not having a printed copy of your boarding pass or other strange rules and hidden fees.
Budget airlines include (but not limited to):
Europe: EasyJet, RyanAir, WOW Air, Wizz Air, Transavia, Vueling
United States: Southwest, JetBlue, Allegiant, Spirit, Virgin
Canada: Sunwing, Air Canada Rouge (not the regular Air Canada), Flair (formerly NewLeaf)