There are plenty of internet Cafes in Pyatigorsk. The easiest ones to access are located at almost every post office and college campuses.  Though post officces are easier to get into, campuses have advantage of English (or somewhat) speaking person if you need help.

You can also try to look around for large signs throughout the city - the word Internet will appear in English and is self-explanatory. Be ware that the connection there may not be as good and prices may vary greately.

There are actually only a few internet cafes, the best one is at the post office (in back) at the corner of Broadway and Mira streets. Prices are ridiculously cheap, about 10 rubles for 1/2 hour. Stay out of the cafes on the side streets, not because they're dangerous or don't work, but the computers are always used by kids who are hooked on online games and buy a whole day of time upfront. Connection speeds are very good anywhere, but Russian internet is a little different then what you may be used to, some websites can't be accessed.