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What does 过 means when it follows a verb?

actually this is the most compliated question you can ask about chinese grammar. you can write a paper about it.

过 is easier to use than 了. not to discourage you but even many "Chinese experts" are not able to use 了 as natural as a native speaker does. the most practical way to learn it is to learn by examples. find as many examples as you can and try to "feel" what it actually means.

I wanted to explain to you theoretically but I find that it takes my grammar book 40 pages to explain 了 and 过. so I think I'll pass.

but intuitively, 过 is more like "have done" and 了 is more like "did". I remember a article that says 80 percent of the time 了 will be translated into simple past tense instead of perfect tense.

"verb + 过" usually has something to do with what you are talking about right now. it not only describs sth that has happend, it relates to now. in contrast, 了 only descripes that "sth happened" or in other words, "the situation changed".

an important and maybe confusing thing to notice is that 过 actually has two functions. it can be an auxiliary as discussed above. it can also work as a complement. when used as a complement, it means you have finished sth., and can usually be replaced with 完. for example,

-你昨天让我看的那本书，我看过了，很有趣。
The book you asked me to read yesterday, I have read it, it's interesting.

here 看过了 can be replaced with 看完了.

compare with
-看一下这本书吧，很有趣。 Read this book, it's quite interesting.
-这本书我看过。 I've read this book before.

in the first example, you can also see the use of 了. let's say you finished the book exactly at 8 o'clock in the morning. then one second before 8, the situation is 没看完 (which means haven't finished yet, and here you can't replace 完 with 过 reversely), one second after 8, the situation becomes 看完了. 了 marks that the situation changes.