Slang terms can be as simple as "Howdy," which is a replacement for "Hi. How are you doing?" Or they can as complex as "Wha'saaaaaaaaap?" which was a popular greeting in the 1990s and early 2000s. Slang terms can cover everything from simple greetings, to abbreviated versions of form or informal words (e.g., y'all is short for you all), to even the numerous variations of common curse words. In order to give your character more life, offer them a few choice slang terms that they may pepper into conversation as the need arises.
Slang phrases, on the other hand, can become either prettier or nastier versions of otherwise common phrases. Some examples might include:
"Amorous Congress" as a replacement for having sex,
"Hot as a half-f@@@ed fox in a forest fire" which would be slang for, well, REALLY HOT.
For breakfast you might order a "Cluck and Grunt" (ham and eggs) and some "Dog Soup" (water).
With a simple search online, you can easily find numerous phrases like these. But it is also really fun to come up with some of your own as well. Your character will thank you, especially when they get into a bar-room brawl and nearly get the "Chicken eggs knocked out of the nest" (teeth knocked out of the mouth).
RPGs can be a ton of fun, and even more so when you begin carrying on conversations and interactions AS your character as opposed to you as the player just describing what your character is saying. Being able to conversate as your character allows you to better communicate with your fellow players and the DM/GM. And depending on what is said by your character and how it is said, you may come up with some VERY interesting interactions and reactions. Give it a try! Your character will thank you.
To get you started, here is a list of 83 Old Slang Phrases for you to try out in your next game session.