Trailers use a very specific type of sound design, and few people even know what those sound effects are even called. Descriptors for trailer sound effects generally include:
Hits & Booms
Hits and Booms are exactly what their name implies. Sound effects that are big hits, booms, impacts, slams, whatever out there is happening suddenly whether it’s a traffic light turning on, someone slamming their fists together, someone landing hard on the ground. There’s a good chance these are getting emphasized with a hit.
Whooshes are for sweeping motions. They are often followed by a hit. Instead of thinking about cuts, think about fades, head turns, motion in general. They can be action-y, they can be flow-y, they can be ethereal or windy.
Rises also typically precede hits. Think of a helicopter slowly powering up, or the sound of a plane as it flies in from the distance then is suddenly very close to you. These build anticipation; when used well they keep you on the edge of your seat to find out what will happen next. You might not realize there’s a rise affecting your mood, but until you hear the end of it you’ll be somewhat unfulfilled.
Drones are for setting the scene or mood. Imagine a desolate apocalyptic scene; what does that sound like? A lot of times, a drone fills in that dead air. Not quite music, and also not just sound design.
NOTE: When you find a sound effects library, check their licensing information, because several trailer sound effects libraries license on a per use basis, and are not free to use willy nilly.
Here are the libraries I currently use for my trailers (royalty free). They range in price from free to a few hundred dollars each, but don't assume that the cheaper ones are low in quality!
Adobe Audition library - Adobe has an amazing FREE sound effects library that includes everything from ambience, cars, weapons, water, impacts etc. Did I mention it's FREE!?
Big Fish Audio Blockbuster Trailer - A great start to your trailer sound effects library, and very affordable!
Big Fish Audio Cinematic Sound Design - Another good starter library
Boom Library - Has a vast selection of sound effects libraries that include a handful specifically for trailer sound design.
Video Copilot MotionPulse - This library has a lot of action movie sound effects, and also some for trailers.
Hollywood Edge Sound Designer's Toolkit 1, 2, 3 - Sound Ideas is a pillar of royalty free sound effects in the industry. Since it's such an institution, a lot of their sounds might sound dated, but it's still a very versatile library.
GDC Audio 2017 Game Audio Bundle - 20+ GB of free sound effects! Royalty free, no attribution required!