If you want an action to happen faster, you use "up". You can also use it if an action has not started yet but you want it to start. The only common phrases I can think of that include "up" are "listen up" (Means "start listening". Sometimes teachers say this to students that are talking instead of listening), "eat up" (begin to eat). What other words are you trying to use it with?

Yeah, the use of the word 'up' is a weird one but it implies that that you need to increase the rate you're doing something. For example, 'hurry up!' means increase the speed you're doing something. 'Speed up!' means to increase your speed. 'Listen up!' means to increase your listening. 'get up' means to quickly stand up. These are commands and are quite rude, so don't use them unless you're talking to someone you are really close with or are giving a command to someone much younger (most commonly a group). An exception is probably 'stand up' because this isn't always a rude command.

In addition, you can use it to say 'I sped up' or 'She told me to speed up'. The opposite would be 'I slowed down' or 'she told me to slow down'. This isn't too informal.

Keep in mind its kind of slang to use 'up'. If you want to be more formal you could say 'Increase your speed', 'Listen closely', or 'Please hurry'