Always book directly with airlines. You'll almost never find a cheaper price from a third party, unless you are a corporate traveler with negotiated agreements through a corporate travel agency. Airlines will often only let the third party who bought the ticket change it, so you have to go through them and often pay a fee if something goes wrong. It's the opposite with hotels. For airlines, use third party sites only to find routes and estimated prices, then see what the cost is to book directly with the airline. Google Flights is usually the best search engine for most people, but don't clickthrough the links to book directly with the airlines. ITA Matrix is the backend to Google Flights and if you're a HNer, you'll appreciate the power interface.

> visit the site twice and they bump up the price

This is a myth, at least for flights. Flight prices normally go up the closer to the date. They almost never go down as you get closer to the date. So there will be many cases where you visit a site at one time and see one price, then if you visit it at a later time, your expectation should be that the price has increased. But the price would have gone up anyway. Also, airlines do use # of tickets already sold to set prices, and so you should expect that as time moves forward, it is likely someone had the same idea of going from city A to city B as you on that same date, and they bought their ticket.

It's the same kind of myth as "I was talking about this thing with my friend and then later saw a targeted ad for it, so they must be listening to my microphone." No, they know you're interested in that product or that there is increased demand for that flight from data collected elsewhere.