Try going and listening to people. It sounds simple, but you can find some dramatic/hilarious moments just by listening.

School, work, home, wherever. Just go and find some people, you don't even have to talk to them. The most important part is just to make sure you get a feel for how they're speaking, what they're speaking about, and if it catches your attention why it's interesting.

I get into the mindset of a particular character and do a pass specifically looking at that character's lines.

I find it helps to have a good ear for different accents, euphemisms, manners of speech. And I think that can be learned by watching movies with real unique characters.

Old movies from the 30s-50s tend to be heavy on dialogue, and provide lots of great example for different kinds of speech. There's the street-smart wiseguy, the country bumpkin, the haughty princess, the down-home matron, and on and on.