Our library is finally re-open again.

“Open again” and “reopen” mean the same thing. They shouldn’t be combined unless you want to emphasize that it’s the reopening that is happening again.

For the last six months I just keept reading fromon my tablet.

Well, I did learn new knowledgethings.

One doesn’t really “learn knowledge.” “Gain knowledge” or “learn new things” should be used.

But sitting in the library reading books is a toovery different experience.

“Too” applied to an adjective has a negative meaning.

Generally, the consonant should be doubled if there is a vowel right before it and the vowel is a short sound (doesn’t sound like the letter name).
“Hop” becomes “hopping.”
“Hope” becomes “hoping.”
“Honk” becomes “honking” because there are two consonants after the last vowel.

Library is for poor mpeople like me.

Enjoy air condition in hot summer.

This is a fragment and needs to be combined with the previous sentence. I don’t think it is here justified for stylistic reasons.

Reading all kinds book and free wi-fi limitless.

Again, this is a fragment.

However, government is still concern ed withe public safety.

“A concerns B” means either that 1) B is worried about A or 2) a question about A involves B.
You can also write “the government concerns itself with public safety.”

The library only open from 1pm to 8pm, Monday to Saturday.

I hope that someday they will open like a -7-11.

It seems better to include “that” if the clause introduced does not start with the subject.