Very interesting story :) and quite advanced language. If you like, though, there are still some things you can improve in the way of writing it. I hope below will be helpful.

I'm going to Slovakia!

When I open this page I usually see self-introducing stories. So I decided to write a story of about -with stories it is usually 'a story about', 'a story of' is more like 'history of' something-  visa receiving process, first in my life.

I was abroad many times. If my mind memory serves me right, it there -with posessives you always use 'there' - was about 15 visas in my old foreign passport. As you may know, Russians have 2 passports, not ID carts. One of them is internal, only for movements -it is better to say 'for moves' or 'travels'- within the country. And the other one -if you use 'other' to refer to the rest of options you should use 'the other', like one passport is for this, the other is for that- is for trips abroad. So, if you want to go abroad, you need to get a foreign passport first and after that you may apply to an embassy for a visa.

I worked as an engineer and trips to different countries were the a part of my work. When I had come to Russia for vacations, I didn't want to go anywhere else -in this sentence it is not clear what you mean: when you went on vacation in Russia you didn't want to go anywhere else? After you had come to Russia for a vacation, you didn't want to go anywhere else- So I spent my time at home.

Not long time ago I retired -retired normally would mean 'permanently leaving any sort of work' eg such as what old people do, you mean 'quit you job'-. I began to miss my trips and spending time in another country -another country or other countries- didn't seems to me as such a bad idea. But I didn't had never applied apply to an embassy for a visa by myself earlier. All I did is was getting ticket and passport from the trips organizing department and go going to the airport -you should use the same tense-.

So. I decided to go to Slovakia this summer by car. But in at Slovakian embassy there (remember - 'there are') are strong rules for tourists. I needed to ask a hotel to send to me the original invitation by via is better here DHL, to take a document about my work and salary, a statement of either about or from bank account. But it is not as difficult to collect all of these documents as it is to make myself to get up at 5 am and go get to the embassy by 7 o'clock, to be the first in front of the door. By the way, embassy is opened opens at 9 am.

I was first. When embassy was opened there were about 50 people after me. And about 10-15 of them were out of turn and come came to me to say that I must to skip them forward let them before me because they were here the day before. It was terrible for me to understand realize -is better here- that you may get up at 5am and be the last in the line. I began to think to not to go abroad. But from  on the other hand (remember - other, the other) I'm tired of working and I needed to go somewhere for a vacations. So, I skiped let 10 people in front of me because there was a list which the security guy composed the day before and they were in this that list. But other 5 people prefered not to go and wait for their turn a queue. I handed over the documents to the embassy and now I'm waiting for a visa. Yahoo!

By the way! Where to go in Slovakia? Which beer you advice advise to taste?