I migrated to Cyprus at the beginning of this year.

This isn't technically wrong, but (at least in American English) it would more normal to say "I immigrated to Cyprus" instead of migrated. While it can be used to talk about people, we mainly use "migrate" to talk about animals. It can be used to talk about people in certain examples (eg "Humans will have to migrate north away from the equator because of climate change.")...but I would either say "immigrate" or "move to" when talking about people in your example. It would also be correct to say "I moved to Cyprus", which is more informal and what most people would say.

PartMany of them migrated from Georgia and Armenia they. They were born in the USSR which is why their generation still knows some Russian.¶
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OR¶
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Many of them migrated from Georgia and Armenia and were born in the USSR which is why their generation still knows some Russian.

There is a big community of Georgians and Armenians they evenat even has some weight in the government.¶
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OR ¶
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There is a big community of Georgians and Armenians and they have some weight in the government .¶
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One Greek taxi driver lifted medrove me (or took me) to an airport, and I discovered that Pontic Greeks can speak Russian too.

Community can be treated either as a singular or plural in this example. If you think of community as a thing, it has weight. If you think of community as a group of people, they have weight. "Who" might be able to be used here too, but one of these would sound better to me.

We were sitting on a bench talking of nothing when Greek senior Greek women stopped and made a compliment in Greek. (or "complimented us/me in Greek")

If you say compliment, it's usually best to define specifically who is being complimented.

I understood that it was thea compliment because, after our confused faces, she pointed at my legs and gave a thumbs up.

She didn't speak English at all, but when she found out that we awere Russians (or "that we were Russian"), she switched and with poor Russian said how much she loves Russian literature and that she could have spoken better in the past but it was difficult to find a Russian book here.

For now, I practice some Greek in Duolingo and learned some greetings and words of gratitude but I'm scared to use them because I feel like an imposter who trying to trick locals with little knowledge.

I migratoved to Cyprus at the beginning of this year.

PartSome of them migrated from Georgia and Armenia, but they were born in the USSR which is why their generation still knows some Russian.

"migrated" is OK here.

There is a big community of Georgians and Armenians theywho even have some weight in the government .¶
One Greek taxi driver lifteddrove me to anthe airport, and I discovered that Pontic Greeks can speak Russian too.

We were sitting on a bench talking of nothing when Greek senioran old Greek women stopped and made a compliment in Greek.

I understood twhat was the compliment was because, after she looked at our confused faces, she pointed at my legs and gave a thumbs up.

She didn't speak English at all, but when she found out we awere Russians, she switched, and with poor Russian said how much she loves Russian literature and that she could have spokenspeak better in the past but it was difficu/\]lt to find a Russian book here.

For now, I'm practiceing some Greek ion Duolingo and learned some greetings and words of gratitude but I'm scared to use them because I feel like an imposter who is trying to trick locals with a little knowledge