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Hi, there.

I faxed the original copy of a prescription to a pharmacy and later I will bring the prescription with me when I pick up the medicine.

Is it Okay to say ," I will bring it (the prescription) in when I pick up the medicine"?

Thanks.
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If this is what you mean then yes it's fine:

I faxed the original copy of a prescription to a pharmacy and later I will bring it with me when I pick up the medicine.

You could also say this, which sounds better to my ear:

I faxed the original copy of a prescription to the pharmacy and later I will take it with me when I pick up the medicine.
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Thanks, MalRey.

A pharmacy means any pharmacy. I didn't mention which pharmacy I was going to; therefore I used "a pharmacy" instead of "the pharmacy" in the context.

'To take it with me' sounds better than 'to bring it with me' in the context. I wonder why? Can you explain?
meantolearnA pharmacy means any pharmacy. I didn't mention which pharmacy I was going to; therefore I used "a pharmacy" instead of "the pharmacy" in the context.
This is indeed the way the use of articles is usually explained but in this case a native speaker will almost always say "the pharmacy" not "a pharmacy." Why? It's just the way we say it, I guess. You will find the same thing with "the hospital" versus "a hospital" whether or not a specific hospital is known native speakers will almost always say "the hospital."

Here is a brief dialog to illustrate:

A: Did you hear that Joe went into the hospital last night?

B: No. Which one, I'd like to send him a card.
meantolearn'To take it with me' sounds better than 'to bring it with me' in the context. I wonder why? Can you explain?
There's no real reason. In part it depends on what part of the US one is in. In some regions "bring" is the most common word, in others "take" is more common. In part it's how the speaker thinks of the action.

There's a very informative discussion of the whole "bring/take" controversy in Merriam-Webster's Dictionary of English Usage. It starts on page 200, here's the link to the Google books version:

http://books.google.com/books?id=2yJusP0vrdgC&printsec=frontcover#v=onepage&q=take&f=false
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Hi, MalRey.

Thanks for the examples you gave. I appreciate them.