I've only been wearing a T-shirt recently, so I feel cold in the morning and evening.

So, after I get off work this afternoon, first I will go to some shops first to buy a long-sleeve T-shirt.

Today is Friday, which means I am going to rest for the following two wo days starting tomorrow / to rest tomorrow and Sundays.

We don’t usually use “the following two days” when the starting/reference point is today. And if we say “for the next two days,” it would usually include now, or at least imply that your rest period was starting now.

My corrections above are two ways of saying what you mean that sound natural to me.

When I woke up at 2pm this afternoon, I immediately prepared / got ready to go to work.

I carried a bag,(,) in which were a few table tennis balls, two table tennis paddles, a bottle of green tea, a towel and a suit of sportssome gym clothes.

After I rode it for a couple of weeks, I added a motor to it, since I often felt tired if I rode it for a long time.

This sentence sounds pretty weird in the sequence of this story—it sounds like it took you 2 weeks to get to work (today)! :-)

It would be less confusing if you put this sentence in parentheses, to show that it’s an aside, not part of the story: (After I had ridden my new (?) bike for a couple of weeks, I added a motor to it, since I often felt tired if I rode it for a long time.)

Another plan after I get off work this afternoon is to [go to some shops to buy | go shopping for] a long-sleeve T-shirt.

I’ve only been wearing a T-shirt recently, so I feel cold in the morning and evening.

suggestion: I’ve been dressing in just a T-shirt lately, so…

I do have a few long-sleeve shirts, but I don’t want to wear them since I think [buttoning / doing] up buttons is a lot of trouble.

So, after I get off work this afternoon, I will (first) go to some shops (first) to buy a long-sleeve T-shirt.

or:
- …I will go directly to some shops…
- …I will first go shopping for a…

Today is Friday, which means I am going to rest for the followingthese next two days.

When I woke up at 2pm this afternoon, I immediately prepared got ready to go to work.

I carried amy bag, in which werehad a few table tennis balls, two table tennis paddles, a bottle of green tea, a towel and a suit of sports clothes (and some gym clothes).

After I get off work this afternoon, I will go to a sportsrecreational center to play table tennis.

After I rodeWhen I first got it, it was fine. But after riding it for a couple of weeks, I added a motor to it, since I often felt tired ifwhen I rode it for a long time.