Suwon, which is located atin the southern part of Seoul, is athe capital city of the Gyunggi province. Suwon, which is located in the southern part of Seoul, is the capital city of the Gyunggi province.

It should be "the capital city" since the province only has one capital city. If it were one of several, you would use "a."

I have been living here since 2010 after joinstarting my job. I have been living here since 2010 after starting my job.

It's more common to say "starting a job."

It is convenient to go to other cities from here, especially, going to go to Seoul, so I can travel every weekend. It is convenient to go to other cities from here, especially to go to Seoul, so I can travel every weekend.

We usually say "go to" a place, with a few exceptions ("go home," "go downstairs," etc.)

Many parks in the city, make me feel relaxed. Many parks in the city make me feel relaxed.

There are big shopping malls and departments as well as small big traditional markets. There are big shopping malls and departments as well as small big traditional markets.

I'm not sure what you mean by "small big traditional markets," but the sentence is fine otherwise.

Suwon is famous aboutfor Bulgogi which was served for the kings of the Choseon dynasty. Suwon is famous for Bulgogi which was served for the kings of the Choseon dynasty.

We always say someone or something is "famous for" something.

Because I can experience all of them, I really like living in this city. Because I can experience all of them, I really like living in this city.