I went out to buy chesse, somesome cheese, bread and vegetables.

or

I went out to buy cheese, bread and vegetables.

After I came backreturned home, the first thinkg I did was to bake a cake for my birthday.

Happy Birthday!

I baked a cake for my birthday.

The previous sentence was a fragment so I made it complete by appending this one.

I celebrated it with my sister.

I also recieived many email messages and carteds from my family and my friends.

Happy bBirthday Courageuse19to me.

We don't really wish our own selves a Happy Birthday. If we did, we probably wouldn't say it to ourselves in third person.

I went out to buy some chesese, some bread, and vegetables.

some can be applied to all items, so you can move it to the front.

After I came back home, the first thinkg I did. was bake a cake for my birthday.

The earlier sentence was not complete (it's an incomplete thought), so I combined it with the following sentence.

I baked a cake for my birthday.

(Can delete this if you combine with the earlier sentence)

I celebrated it with my sister.

Tense issue (keep everything in the past)

I also recieived many emails and carteds from my family and my friends.

Spelling issues

Thisat day iwas the happiest day of my life.

I went out to buy chesese, some bread, and vegetables.

After I came back home, the first think I did. was bake a cake for my birthday.

I also recieived many emails and carteds from my family and my friends.

This day is the happiest day of my life.

I went out to buy chesse, some bread and vegtables (today?).

The sentence feels unfinished without saying when you bought these items.

AfterWhen I came back home, the first thinkg I did. was

Or "When I got back home"
"After I got (back) home"

I bakeding a cake for my birthday.

continuing on from the last sentence

I celebrated it with my sister.

I also recieved many emails and carteds from my family and my friends.

"my family" implies that they are also your friends so you do not need to repeat it

This day is the happibest day of my life.

"Happiest" is not wrong but "best" sounds and suits better here