Today's Events.

Good afternoon.
Today, I went outside to look for stray cats from 2:00 to 5:00 in the morning.
Afterwards, I went home and surfed the internet until 2:00 in the afternoon.
And I tried to go to the cats cafe but I changed my mind and went to the library instead.
Thank you for waiting.

Your last line "thank you for waiting" is correct English, but it doesn't really make sense here. In your Japanese you use the verb for "to look" (見る) , so I think "Thank you for looking" is probably what you wanted to say. 

Also, your words for describing time are correct, you just have them out of order. In English we put the time in hours first and then the time of day later. You can also say 2:00 to 5:00 AM (morning) and 2:00 PM (afternoon)

Today's evens.

Good afternoon.
Today,I went outside to look for stray cats from the morning 2:00am to 5:00am.
After, I went home and play net surfing surfed the net the afternoon until 2:00pm.
And I try tried to went go to the cats cafe but instead changed my mind and I went to a the library because I change my mind.
Thank you for waiting.
こんにちは。

私は今日、午前2時に家を出て5時まで野良猫を探し歩いていました。
その後家に帰り午後1時までネットをして時間を潰しました。
そして2時に猫カフェに行こうと家をでたのですが
気が変わり図書館によりました。
見てくれてありがとう。