Gyms DuringOn Saturdays.

"During" is correct in this context, but "On" is much more natural.

Today I was havinghad a little cold.

This is a tricky and maybe slightly idiomatic structure. I'm not sure why we don't use past continuous here, but we'd always say "I had a cold" and never "I was having a cold" in my opinion.

Gyms duringon Saturdays are very calm and quiet.

Just replacing "during" with "on" again here.

I wish my gym were calm and quiet on Saturday!

I enjoys working out on weekends because I don't have to wait for my turn to use the gym equipments and machines.

"Equipment" is always singular, never plural.

But, due to my cold and weakness I didn't worked out.

"...due to cold" sounds a bit like it's due to cold weather, where as "...due to my cold" makes it clear that it's referring to your illness. (I hope you're feeling better very soon, by the way!)

I didn't work out --> The past tense is already established with "did" instead of "do" in this sentence, so work does not take the past participle (worked).

I just missed the chance tof havinge a great weekend.

Today I was having a little cold.

Use the word "had" instead of "was having"

But, due to cold and weakness I didn't worked out.

Include a comma after "weakness"-- this is called an appositive sentence. Also add "the" in front of "cold." Without including "the" it sounds you are saying because of the cold weather, you didn't workout. In this case, it is because you felt sick, so "the cold" would be correct.

Today I was having a little cold.

I "enjoys" working out on the weekends because I don't have to wait for my turn"long" to use the gym equipments . (adding "and machines." isn't necessary.)

But, due to cold and weaknesshaving a cold and feeling weak, I didn't worked out.

This sounds more natural.

I just "missed" the chance of having "a" great weekend.

You're talking about the past so "miss" becomes "missed" and adding "a" helps it sound fluid.