You're probably taking a boat at some point along your journey, but depending on your starting point, the trip will differ in various ways.

The first Varangians were soldiers from the kingdom of the Kievan Rus people—the ruling elite were ethno-linguistically Scandinavians who had settled in Eastern Europe during the Viking Age, and the general populace were of more Slavic origins. They made their way down the Dnieper to the Black Sea and then to Constantinople and made contact with the Byzantines (as explained here). The first contact was violent, but then the Byzantines started to hire them as mercenaries to fight in various conflicts. Some Rus stayed behind and formed the core of the guard.

As the Varangian Guard grew, it started to attract other Germanic peoples from further West, especially in Scandinavia and the British Isles. In the 11th century, the Guard saw a major influx of Saxons and Danes who arrived via ships that had sailed into the Mediterranean.

In other words, to get from your homeland to Constantinople, you're probably spending some time on a boat. How much time depends on whether you're starting out in Eastern Europe (in which case you might sail along the Dnieper and then through the Black Sea to get to Constantinople) or in Western or Northern Europe (in which case you might sail along the Western European Coast and into the Mediterranean to get there).

Once you're there, the Varangians are actively recruiting people like you (Germanic warrior types) and probably have connections to people you know, so you form up with those connections/those recruiters, who are ethno-linguistically similar to yourself, or at least, more similar to you than the rest of the general populace in the area is, and you join the payroll.