Today, it is rainyed all day and I didn't go out.

The weather forecast says it will be rainy and cloudy next week, so and I'm fed up with it.

I don't dislike rain per se but I dislike that it causes high humidity and makes me being short ofhave to postpone my outdoor exercise.

I assumed you meant that the rain caused you not to be able to exercise outdoors as much, so that's why I wrote it like this!

It i's Rainy

This is correct but is very formal. Formal talk like this is reserved for legal or technical language, and can make you sound defensive. 'It's rainy' sounds more natural.

Today, it iIt was rainying all day today and I didn't go out.

Alternatively, 'Today it was raining all day and I didn't go out,' which is a little more stiff but entirely acceptable. When it comes to is vs was: if it encompasses the entirety of a thing, it is past tense. I'm sorry if that's a bad explanation, I can't think of a better way to say it.

The weatherI'm fed up with the weather, since the forecast says it will be rainy and cloudy all next week, so I'm fed up with it.

You are the subject because this is about your feelings about the weather, so you take priority in the sentence.

I don't dislike rain per se but dislike that it causes high humidity and makes me beingfall short of my exercise goals.

Being implies it started that way. If you don't have exercise goals but just exercise more when its not raining, you would just say 'and it makes exercising hard.'

Additionally, I canno't dry my clothes without going to a laundromat, which costs time and money.

Correct but uncontracted 'cannot' is very formal.