Some are used slightly differently. "Skim" usually means to read something that is important but quickly, without reading all of the words: skim a contract, skim an essay, skim an academic book. Leaf: you usually "leaf through", something large slowly, without reading all parts of it: leaf through a newspaper. Flick: you "flick through" something unimportant without reading very much, and usually just look at the pictures. So you typically flick through a magazine. I have never heard anybody say "thumb through". There is browse through: this could be a text of any type which you go through leisurely and slowly.

I would use 'Flick through' if it were something i really didn't care about reading, like if you were waiting in a reception area for a doctors appointment and there just happened to be some magazines there you were reading but only because you had nothing else to do. You might however have 'had a quick flick' through a magazine but in that case its usually because of a lack of time. Like, i was rushing between meetings so i quickly flicked through the report but the assumption is that you haven't necessarily seen it all. 'Skimmed' to be me is interchangeable for the two examples above. I also agree with torusan, i would more likely say "I 'quickly looked through' the car magazine in the grocery store" Thumbed and Leafed i would not use. I would understand what is meant if you said it but its a rather odd or outdated way of saying it.

"Flick" is *really* brief, like flicking a cigarette. For things like books or magazines, I would tend to say "flipped through" (instead of "flicked"), "skimmed through" or "thumbed through". Another possibility would be "I took a quick glance" or "I quickly scanned through it".