Do you prefer to stay atin one place or move around?

I think 'stay in' sounds slightly better but I'm not sure why.

'Live in one place' may be better though. 'Do you prefer to stay at/in one place' might be interpreted as 'Do you prefer to sit down, or walk around?'

Just as some people enjoy living in one place, others think that live ing in several different places is a better option.

First off, I would rather live in many places than only stay at one because livingwhen you live in different places, you have the chance tof meeting new people, and will probably make new friends.

It's 'chance to __', not 'chance of __'.

I feel this way since you will surely will bebe in contact with them.

This makes sense but it sounds a bit confusing and awkward. 'Be in contact' is a bit formal.

It might be better to just say 'If you see other people often, you will eventually become friends.'

For instance, maybe you will come across other people when you take out the trash or go to work, maybe you will come across with them.

This is correct (except it should be 'come across them') but it sounds a bit awkward.

Consequently, you get to madke some new friends as you moved around.

You suddenly changed tenses here. You went from present tense to past tense.

Secondly, meeting new people is not the uniqueonly benefit of moving around, h. Having new experiences is another reason why I would decide living in other placto live somewhere else.

'unique benefit' implies 'Moving around is not the only way to meet people'
'only benefit' implies 'Meeting new people is not the only benefit of moving around'.
So I think 'only benefit' is better.

For example, the environment of your new neighborhood.

This is a bit awkward on its own. What is good about the environment of your new neighborhood? Maybe you could write 'For example, being able to enjoy life in your new neighbourhood.'

If you live in a newmove house, you might be able to enjoy new events, fairs, or even join sports teams.

'Live in a new house' emphasizes the 'new house' part. (And so it implies that you wouldn't be able to have fun if you lived in a 50-year-old house). 'If you move house' works better.

Thise following experience is a compelling illustration of this.

'The following experience' makes it clearer to the reader that the next sentence is where you talk about this experience.
Otherwise this sentence is fine, but 'compelling illustration' sounds a bit overly formal. 'The following experience is a good example of this.' would also work

When I started studying for my career, I lived in othera different city.

At that placWhile I was there, I joined a football team and every weekend I played football in a tournament.

Finally, the opportunity tof livinge in different places not only helps you havinge new experiences, but also broadens your mindset.

I reckon people may learn new things if they live in different placessomewhere new.

Three years ago, I moved to Taiwan (Asia) forto learning Chinese.

There, I couldwas able to broaden my horizons because I lived in other country and understood how Taiwanese people thinks.

'Was able' is better here.
'Could broaden my horizons' = 'It was possible for me to broaden my horizons (but I didn't necessarily broaden them)'
'Was able to broaden my horizons' = 'It was possible for me to broaden my horizons (and I did)'

Although at the beginning was very hard, I couldwas able to adapt to the new environment.

Alternatively you could say: 'Although at the beginning it was very hard'

Meeting new people, having new experiences, and broadening your mindset are only a few of them.