The Beginning of Autumn this year iAutumn beings on the seventh of August, which means this year which will be the start of fall. Autumn beings on the seventh of August this year which will be the start of fall.

Just made it sound more natural :)

The Bbeginning of Aautumn this year is on the seventh of August 7th, which means the start of fall. The beginning of autumn this year is on August 7th, which means the start of fall.

Unless you consider the beginning of autumn to be really special, I'd keep things lowercase. Also, writing out "the seventh of August" isn't wrong, but I find it simpler to just write "August 7th" instead.

The Bbeginning of Aautumn this year is on the seventh of August, which means the start of fall. The beginning of autumn this year is on the seventh of August.

Before thaten, it was very hot. Before then, it was very hot.

People needed to turn on their air conditionersing all day. People needed to turn on their air conditioning all day.

Not incorrect but this is a more common way to say it :)

After thatNow, the weather is getting cooler and people feel very pleasantare feeling better because it is not hot anymore. Now, the weather is getting cooler and people are feeling better because it is not hot anymore.

After that, the weather is getting cooler, and people feel very pleasant because it is not hot anymore. After that, the weather is getting cooler, and people feel very pleasant because it is not hot anymore.

Alternatively, you could delete the "and" in order to make this sentence into two shorter ones.

After that, the weather is getting cooler and people feel very pleasantbetter because it is (not as) hot anymore. After that, the weather is getting cooler and people feel better because it is (not as) hot anymore.