(sobre hoje)about today

Eu gostaria que você corrigisse a minha escrita em português. Obrigada!

ou

Alguém poderia corrigir o meu texto por favor?

Hoje estava muita neve hoje nevando muito. (today was snowing really hard.)
Estava muito frio, mas e a neve emphiliada acumulou (ou empilhou) na rua. (it was no so cold but the snow piled up on the street)
Alguns ônibus não correram *rodaram por causa da neve pesada. (Some buses didn't run due to the snow)
A neve é bonita mas afeta o transporte público. (the snow is beautiful but it affects public transportations)

* No Brasil é comum falarmos "Hoje os ônibus não vão rodar" com o sentido de que os ônibus não vão sair das garagens hoje.

(sobre hoje)about today

Sobre hoje

Não estava muito frio, mas a neve empilhava na rua.

Alguns ônibus não correram trafegaram por causa da neve pesada. 
A neve é bonita mas afeta o transporte público. 

A neve é bonito mas afeta o transporte público. (the snow is beautiful but it affects public transportations)

(sobre hoje)about today

Quick tip: Always remember that "thank you" in portuguese has gender (so as everything, hahah). Boys say "obrigadO" and girls say "obrigadA")

Hoje estava muita neve hoje. (today was snowing really hard.)
Hoje estava nevando muito/demais/bastante. (I've included some synonyms)

You have used "snowing", then you should use the portuguese gerund "nevando". If it was raining (chuva) you would use "chovendo".
"Muito" is often used after an adjective or noun to express quantity or intensity. It means "very", "much", "many" or "a lot".
Examples: very good; very cold. | much money (uncountable); many apples (countable: "muitos"). | A lot of friends (muitos);
In this particular case you should use like you have used "really hard" to mean intensity.
Example: Today I worked really hard. = Hoje eu trabalhei muito. | They are studying really hard. = Eles estão estudando muito.

Estava muito frio, mas a neve emphiliada na rua. (it was no so cold but the snow piled up on the street)
Não estava muito frio, mas a neve empilhou na rua. (You said between parenthesis that it was NOT too cold, but you wrote the opposite).

Alguns ônibus não correram por causa da neve pesada. (Some buses didn't run due to the snow)
Alguns ônibus não rodaram/circularam por causa da neve forte.

Rodaram = I see some people have explained you above.
When you are speaking about "heavy rain", "heavy snow" or "heavy sun", it's translated as "forte" (strong).

A neve é bonito mas afeta o transporte público. (the snow is beautiful but it affects public transportations)
It's all correct but the gender (bonita). Good job!

Good job! You are doing pretty well. I hope I have helped.

Oi, Kana!

(sobre hoje)about today

You're welcome! =)

Hoje estava muita neve nevando demais hoje. (today was snowing really hard.)
[Try to use some connectives to link your phrases... Look how one word can change your text] Também estava muito frio, mas a neve acumulou [We use the word "empilhada" more for blocs, or construction materials] na rua. (it was no so cold but the snow piled up on the street)
Alguns ônibus não correram por causa da [more usual]/ devido à [more formal] neve pesada. (Some buses didn't run due to the snow)
A neve [You can substitute for "Ela" - pronoun - in order to not repeat many times the same word in the same context.] é bonitA [Look! You wrote "A neve", so it's feminine, consequently the adjective "bonita" should be feminine too], [Before negative connective as "mas", "porém", "entretanto", "todavia" - everything means "but", "however" - you have to put a comma] mas afeta o transporte público. (the snow is beautiful but it affects public transportations)

Very good job!! =) Keep practicing!