Which is correct or more common when talking about medicine: "buy drugs" or "buy medicine"?

"Drug store" (or as it is commonly spelt, drugstore) is a term that is local to the US (and possibly to some parts of Canada). The rest of the "English-speaking" world mainly uses chemist or pharmacy, with medicals, medical store, and druggists (usually for stockists) seen in places in the Indian subcontinent as well as in Africa.

It is very rare to hear, "I'm going to the [store] to buy drugs". The words medication, medicine, and pills are far more common in this context. You might hear the word drug used in the context of "the latest drug in the market". But in day-to-day usage, it is mostly used to refer to narcotics.

The latter. 'Buying drugs' would imply illegal drugs, like heroin or cocaine.

The former could be used facetiously:

Colleague: What are you doing here? I thought you had the flu.

Me: Yeah well, I'm on a ton of drugs. I'll survive.

As has been mentioned by others, the term drugstore is American English. See: http://dictionary.cambridge.org/dictionary/british/drugstore?q=drugstore

In England and the rest of the UK, it is usually called a chemist. See: http://dictionary.cambridge.org/dictionary/british/chemist_2 and http://www.ldoceonline.com/dictionary/chemist

The word pharmacy is also used sometimes in the UK. See: http://www.ldoceonline.com/dictionary/pharmacy and http://dictionary.cambridge.org/dictionary/british/pharmacy?q=pharmacy

The links to www.ldoceonline.com are particularly useful because they have an explanation of how to use these words.

Which term you use will depend on which style of English you are using. Use drug store if you are speaking American English. Use chemist if you are speaking British English.

When talking about buying medicine, which of these sentences is more correct or more commonly used:

"go to the drug store to buy drugs" "go to the drug store to buy medicine"

The second one, which would be "go to the drug store/chemist (depending on which English you use) to buy medicine". Talking about buying drugs, is normally a reference to buying illegal drugs (not medicines).

As several have pointed out, in U.S. usage, buying drugs often has the connotation of buying illegal drugs. Probably most users would say to buy medicine, but both are heard.

Interestingly, when referring to the developers and sellers of such products, the companies are called either pharmaceutical companies or drug companies, but not medicine companies.