I’m from Ukraine but I’ve moved into Ireland.

"I've moved in Ireland" -> You've moved from one part of Ireland to another part of Ireland
"I've moved to Ireland" -> You've moved from somewhere that is not Ireland to Ireland.

I was studying iat an Irish school last year.

And dDuring the year I had “work experience” twice.

It's more natural not to start a sentence with And.

The first job I have tried iwas working in the gym.

My job was cleaning all the doors, windows, machines and equipments.

"equipment" is uncountable - if you wanted a countable term, you'd use (a piece / pieces) of equipment.

Don't forget your spaces after punctuation.

Also I was cleaning swimming pools and it was awful.Not only It wasn't just cleaning the swimming pools that was awful and, but all of my job.

While all the other staff did nothing I was doing all of the work.

I’m really upset that I spent my time forthere, it was the worst week of my life.

I’m from Ukraine but I’ve moved into Ireland.

The phrase is "move to (somewhere)".

I was studying in an Irish school last year.

And during the year I had “work experience” twice.

You can consider merging this sentence with the previous one: "I was studying in an Irish school last year, and during the year I had 'work experience' twice."

The first job I have tried iwas working in/at the gym.

(1) "Have tried" is what is known as the present perfect tense. It's used when the past time frame of the event is not specified. However, since you did specify the past time frame here ("last year"), the simple past will be more natural.

(2) Both "working in" and "working at" are fine here; they're both used to specify the location of work. But be aware that "work at" may also place a small emphasis on the specific company whom you're working for, in addition to the location you work in, which here you may want.

My job was to cleaning all the doors, windows, machines and equipments.

(1) We'd typically use "my job was to..." The same applies for other similar constructions, such as "my goal was to...", "my task was to...", "my priority was to..." etc.
(2) "Equipment" is generally an uncountable noun.

Also I was cleaning swimming pools and it was awful.Not only cleaning swimming pools was awful and all of my job Really, the whole job was awful.

I'm just going to repeat another user's corrections for the last part since it works.

When I was working there I hated every day and eachvery person who told me about my “bad” completed work.

"Each person" is probably grammatical, but "every person" might sound more natural, especially given that you'd also used "every" in "every day". Alternatively: "I hated everybody" or "I hated everyone".

While allthe rest of the staff did nothing I was doing all of the work.

Alternatively: "I was doing all the work while the rest of the staff did nothing." or "I did all the work while the rest of the staff did nothing." There are multiple ways to convey the same thing.

I’m really upset that I spentwasted my time foron the worst week of my life.

(1) "Wasted my time" might be more appropriate here, given the negative attitude you're trying to convey.
(2) Alternatively: "I'm really upset that I spent time working there just for it to be the worst week of my life."

I’'m from Ukraine but I’ve moved into Ireland.

I was studying istudied at an Irish school last year.

Aand, during the year, I had “"work experience”" twice.

The first job I have tried id was working inat the gym.

My job was to cleaning all the doors, windows, machines and equipments.

Also I was cleaninged the swimming pools and it was awful.Not only cleaning swimming pools was awful and all of my job Really, the whole job was awful.

"Also, I had to clean the swimming pools ..." sounds a bit more natural.
"Really/Honestly/to be honest" gives the feeling of "not only all of that, but also...". It's like a clarification and I think this was your intention.

When I wasile working there, I hated every day and each pversyone who told me about my “"bad” completed" work.

More natural: "and everyone who complained to me about ..."

While allThe whole staff did nothing I was doingwhile I did all of the work.

I’m'm still really upset that I spent my time for the worstwasted a whole week of my life there.

When complaining about time spent, it's very common to use wasted.
"I wasted six hours trying to debug that code"
"She wasted 6 years of her life with that deadbeat boyfriend"
"I'm worried we will waste too much time on this. Let's do something else"