It would depend on who you were, and who you were looking for. Travel was of course slow and uncertain, and even those with access to the imperial fast post (more on that in a moment) would have had to wait weeks for replies from a distant correspondent. Once you reached the city where you were trying to find someone, you would have no choice but to ask around (a much more productive method in the face-to-face societies of antiquity than it would be in a modern city) until you found your friend.

Since such generalizations are boring, let's consider four scenarios: a Roman governor in Asia Minor (Turkey) sending a letter to the emperor back in Rome, a Roman expatriate living in Greece visiting a friend in Rome, a merchant from Hierapolis sailing into Ostia (Rome's port), and an apostle from Tarsus looking for the leaders of the Jewish community in Corinth.

Scenario 1 (from Pliny the Younger, Letter 10.96): You are a dutiful Roman governor, out managing restive Greeklings in the province of Bithynia-Pontus (now northwestern Turkey). You have encountered some of those seditious cultists who call themselves Christians; and although many of these criminals swear that they have lapsed and are loyal Roman citizens, a few persist in their error. Being a cautious man, well aware that any over-zealousness on your part might be viewed with disfavor back in Rome, you write to the emperor for instruction on how to proceed. And since you are a governor, you have access to the couriers of the cursus publicus - men with official passes who ride fast horses and are authorized to requisition transport. Finding the emperor (in this case Trajan) is easy: he is currently in Rome. Even when he's on campaign ("Rome" as the saying goes "is where the emperor is"), his whereabouts are always known. Trajan's reply to Pliny also uses the fast post - and Pliny is also easy to find, since (as governor), he is following his regular assize circuit, and is likely to be found in one of a half-dozen or so cities.

Scenario 2 (from Cicero, Letter to Atticus 1.4): You are a rich and remarkably apolitical Roman gentleman, thinking about returning to Italy after a very long sojourn in Greece. You have been making vague promises to this effect to your good friend Cicero, who notes rather peevishly that you're late:

"You keep on making me expect you again and again. Only the other day, when I thought you on the point of arriving, I was suddenly put off by you till Quintilis (July). Now, however, I do think that you should come at the time you mention if you possibly can...."

When you actually do set out, you will send a trusted slave with a message to your friend, who will thus know when and where to expect you.

Scenario 3 (from the epitaph of Titus Flavius Zeuxis): you are a wealthy merchant from Hierapolis (in modern Turkey), making your 36th and final voyage to Rome. For 36 years, you have sailed around the Greek peninsula and into Rome's great port at Ostia. After so many years, of course, you have many contacts in Italy, and you speak Latin reasonably well. But if you hadn't arranged a special meeting place, you will simply go to Ostia's Forum of the Corporations (or whatever its ancient name was), where you will find the offices of about 40 guilds. The men in these offices will be able to direct you to your former contacts and every other seller in town.

Scenario 4 (from Acts 18.1-5): you are a stooped and contrarian tentmaker with a gospel to spread. Having spent some time in Athens, you decide to travel to the larger commercial city of Corinth, a few days south. You know something about the Jewish community in Corinth, and may have the names of some contacts; but it is possible that you just roll into town and start asking around until you find someone to stay with - in this case, a fellow tentmaker named Aquila. Much evangelizing fellows.