You picked a strange time to move! Around 500 BC, Lesbos and the other Ionian islands were under Persian control. Athens wasn't yet openly hostile to Persia (and the Persians even had reason to think, thanks to Athens' unratified submission in 507 BC, that they had subjected Athens too), but even so there may have been less open relations between the two communities than in the half-century that followed. But not to fret: the Athenians would have had little reason to reject a skilled worker who came from their own supposed foundations in the east and who spoke a dialect similar to their own.

Now, we know much more about the kind of things an immigrant would encounter at Athens in 400 BC than we do in 500 BC. A lot of later institutions and infrastructure wouldn't have been in place yet. For one thing, in 500 BC the Peiraieus - the famous harbour city of Athens - hadn't been built yet, and your way to the city would be inland from the shallow beach at Phaleron. For another, a lot of the officials and formalities that you would encounter might have been different than they are at the time when we actually have evidence. Bear this in mind - I will have to reach for later evidence which may not actually be valid for such an early time.

In any case, your first task when you arrived at the city would be to find your proxenos. This was the local resident who was designated to act as a "friend," host, and first point of contact for all envoys and travellers from a particular community. There were only a few poleis on Lesbos and the odds are that you are from Mytilene, the largest community on Lesbos, so you'd need to find the proxenos of the Mytileneans. If you are from Methymna instead, find the guy who does the thing for Methymna. It's pretty straightforward. The proxenos should be able to tell you what to do (though his main business is taking care of high-status visitors, so he probably won't have that much time for a simple craftsman like you).

In later times the main thing would be to find the polemarch, the magistrate responsible for migrant affairs. He will get you registered as a metoikos (metic), or resident foreigner. This was a specific subclass of the Athenian population - freeborn, but below citizen status. As a metic you were liable to an annual poll tax and military service whenever the levy was called up. In return, you would be free to live and work in Athens, and in 500 BC you were even still free to marry an Athenian citizen woman if you wanted, in which case your children would probably be counted as Athenian citizens (this kind of marriage would become illegal in 451 BC). You would not have any political rights, though, and you were not allowed to represent yourself in court.

(It's not clear if the polemarch was already the point man for registering metics at this early stage; his initial role, as his title suggests, was to command the army. This function disappears soon after 490 BC. Either way, the polemarch is one of the nine archons, the highest magistrates of the city, so be polite.)

It's also my sad duty to crush your dreams: as a metic you were forbidden to buy land or own property. Greek communities jealously guarded the landed property on which enrollment in the highest status classes would often be based; they also recognised that rents on land and housing were important sources of income for citizens and the state treasury. So you'll be stuck renting a place for the rest of your life. But that will also help you to establish yourself, since you're likely to end up living in a residential block alongside other skilled migrant workers. They will be able to help you get the connections to find work.

From then on, it'll be a daily grind in a society where you are systematically kept in a subordinate position. Don't get me wrong - it's quite possible for a metic to make a fortune, and even to live forever in the memory of ancient historians, like the metic orator Lysias or the metic philosopher Aristotle. You could have chosen a worse time to move to Athens: in less than 20 years they will begin massively expanding their navy and building the dockyards at Peiraieus, so there will be no shortage of work for a skilled carpenter (just make sure you're out of town when the Persians burn the whole place to the ground in 480 BC). But you will never cast a vote in the assembly or serve on a jury or be counted as an Athenian. Unless Themistokles enfranchises you (it's debated). Still, you got here early enough to hope that your children might.