"Away" has the meaning of not here, or sometimes, not there. So for example, a sound that 'fades away' gets softer and leaves here. "Take away" means to take something and move it to "not there". For example, "My dog was chewing on my shoes, so I took them away," meaning I took them AND they are not in his location anymore.
Other combinations, like the phrase "right away" (meaning as soon as possible) are just idiomatic and should be remembered. Unfortunately, I don't think there's a trick for these. Hope that helps!