1. We Don't Introduce Ourselves with Меня зовут (My name is) or Я - (I am)
“How do you do it then?” you may ask. Well, by simply saying our names. It looks like this: My friend wants me to meet her new boyfriend. She says he will be waiting for us in a cafe with a couple of his friends. When we arrive, my friend waves her hand at her boyfriend so that I understand who the guy I came to see is and says, Это Ваня (This is Vanya).
And then it's my turn to introduce myself. What do I do? I shake hands and say my name, Настя. He says, Ваня. And then I do the same thing with the others. We simply say our own names and shake hands. In a more formal situation I would use my full name: Анастасия (Anastasia). We actually say Меня зовут Анастасия when we introduce ourselves in front of a big audience, in the first entry of a blog, or in a presentation video.
2. We Don't Say Where We Are From By Saying Я - русский (I'm Russian)
You would hear “I'm Russian” mainly followed by “and so…” For example, “I'm Russian and so I'm fine when it's -20 degrees.” Or, “I'm Russian and so I can eat ice cream when it's -20 degrees,” etc. To say where you are from in Russian you have to use a structure similar as in English: I'm from... We say, Я из Росии.
3. We Don't Only Use Future Tense to Talk About Future Events
In the same way English speakers use the present continuous tense to talk about personal schedules, Russian speakers use the present tense. You can use present tense to talk about future events, and most importantly, sometimes with no distinct difference from the future tense. Why do you need to know this? Because future tense in Russian is a bit more complicated than present tense. That said, you shouldn't deprive yourself of talking about the future just because you're a beginner.
It's now February. And if I have just booked my flight to Athens I can say:
Я еду в Афины в Августе (I'm going to Athens in August).
It's morning and I'm talking with my boyfriend about my plans for today:
В 12 часов я иду на работу. В 15 я встречаюсь с подругой. В 18 я ужинаю с мамой. В 20 я еду домой.
At 12 p.m. I'm going to work. At 3 p.m. I'm seeing a friend. At 6 p.m. I'm having dinner with my mom. At 8 p.m. I'm going home.
Иду, встречаюсь, ужинаю, and еду are all verbs in present tense referring to the future.
4. We Don't Only Use Past Tense to Talk About the Past
Now, if you understand why we use present tense to describe future events by comparing it with present simple and present continuous for timetables and personal schedules, this may surprise you a bit. We do also use the present tense to talk about the past. The way we use it is a bit similar to the way native English speakers use present simple to tell jokes:
An electron and a positron go into a bar. Positron: "You're round." Electron: "Are you sure?" Positron: "I'm positive."
We use present tense when talking about the past to make a story livelier.
Сижу я вчера в своей комнате, читаю книгу. Вдруг слышу хлопок, и свет выключается.
So I'm sitting in my room yesterday, reading a book. Suddenly, I hear a bang and the lights go out.
5. We Don't Start Using ты (you informal singular) Immediately
There are a lot of people who I have known personally for a long time and see every week but still address them as вы (you formal singular) because I'm a polite person and, well, we haven't become friends. In eleventh grade I moved from a suburban school to a central one. There everything changed for me. Our teachers addressed us as вы (you formal singular). That was a kind of respect I had never felt before. Strangers calling you ты (you informal singular) just shows their lack of manners.
Before we start to address a person as ты we usually ask how they feel about it. We have a common phrase, “Let's start calling each other ты,” which is much shorter in Russian: перейдем на ты or давай на ты. However, there are a lot of people who think addressing your peers as вы sounds pretentious. So my advice is this: act according to the situation. I personally address all people outside my circle of friends as вы, and so do most polite people in Russia.