For instance if someone says, "Suggest me some books to read."

That doesn't sound correct to me at all. But I cannot find any information as to whether it is correct English usage or not.

Perhaps:

"Suggest some books for me to read." (A "please" beforehand wouldn't hurt that sentence.)

or

"Suggest for me some books to read."

Any help would be appreciated. Thank you so much.
This is apparently okay. This would be similar to other constructions of the same pattern:

Send me an email outlining your request.

Fill me this prescription fast. I'm in a rush.

Give me some of that.

Do me a favor, will you.

Suggest me a title for this article I'm writing, will you.
"Suggest me some books to read." That doesn't sound correct to me at all. That's because it's not correct.

"Suggest some books for me to read." This is how to say it.

"Suggest for me some books to read." No. You don't want anything between the verb and its object. It's not English to do that.

The verb 'recommend' follows the same grammatical pattern.

Use the search box above to find out more about 'suggest' and 'recommend'. Here are a few links to get you started.

See The difference of " recommend and suggest"

See suggest

See Suggest

CJ
Students: Are you brave enough to let our tutors analyse your pronunciation?

You are right.

"Suggest me..." is not correct.

"Suggest" is different from other verbs in this regard.

Please suggest some books to read....