Ho un forte raffreddore, quindi oggi resto in casa.

Because in Italian, avere is used in many expressions where English also uses to have, including to have a cold.

• Ho il raffreddore = I have a cold
• Ho un forte raffreddore = I have a bad/severe cold

So ho is the natural choice here, not sono.

Here forte means strong, but in natural English this is usually translated as bad or severe.

So:

• un forte raffreddore = a bad cold / a severe cold

It does not mean the cold is literally physically strong in some strange way; it is just the normal Italian way to intensify the illness.

Both are possible, but they mean slightly different things.

• Ho il raffreddore = I have a cold
• Ho un forte raffreddore = I have a bad cold

With il raffreddore, you are naming the condition in a general way.
With un forte raffreddore, you are describing one case of a cold, and adding the adjective forte.

So the indefinite article un fits naturally when the speaker is characterizing it.

Italian often drops subject pronouns because the verb ending already shows who the subject is.

• Ho = I have
• Resto = I stay / I’m staying

So io is unnecessary unless you want emphasis or contrast.

For example:

• Io resto in casa, ma tu esci. = I’m staying home, but you’re going out.

In your sentence, plain ho and resto are completely normal.

Quindi means so, therefore, or thus.

In this sentence, it links the cause and the result:

• Ho un forte raffreddore = I have a bad cold
• quindi oggi resto in casa = so today I’m staying home

It is a very common connector in Italian.

It is very natural here because quindi introduces the consequence of the first clause.

• Ho un forte raffreddore, quindi oggi resto in casa.

The comma helps separate the two ideas clearly. In informal writing, punctuation can vary, but this version is standard and clear.

Resto comes from restare, which means to stay or to remain. It works very well for staying at home / staying indoors.

• restare in casa = to stay in the house / stay indoors

You may also hear:

• sto a casa = I’m staying home / I’m at home

So resto is not the only possible verb, but it is a very natural one here.

Yes. Rimango in casa is also correct.

Both restare and rimanere can mean to stay / remain. In many everyday contexts, they are very close in meaning.

• Oggi resto in casa.
• Oggi rimango in casa.

Both are fine.
For a learner, restare in casa is a very useful and common pattern.

This is a very common question, because both can relate to home, but they are not always identical.

• a casa often means at home / home
• in casa often emphasizes being inside the house, indoors, or in the home

So:

• Oggi resto a casa = Today I’m staying home
• Oggi resto in casa = Today I’m staying in the house / indoors

In your sentence, in casa slightly highlights the idea of not going out.

Italian adverb placement is flexible, and oggi can appear in different places.

All of these are possible:

• Oggi resto in casa.
• Resto in casa oggi.
• Resto oggi in casa.

The version in your sentence, quindi oggi resto in casa, sounds very natural because oggi clearly sets the time before the action.

In Italian, the present tense is often used for current decisions, plans, and near-future actions.

• Oggi resto in casa. = Today I’m staying home.

This is completely normal. English does something similar with I’m staying home today.

So even though it refers to what the speaker will do for the rest of the day, the present tense is the natural choice.

Yes. Raffreddore usually refers to a cold as an illness, especially the common cold.

Examples:

• Ho il raffreddore. = I have a cold.
• Mi è venuto il raffreddore. = I came down with a cold.

It is a very common everyday word.

In many contexts, yes, especially in spoken Italian, where it can mean I’m staying in or I’m staying at home today.

However, resto in casa sounds a bit more explicitly like I am remaining/staying indoors because of a reason, such as being sick.

So:

• Oggi sto in casa = acceptable, conversational
• Oggi resto in casa = very natural, slightly clearer for staying in

It is neutral and completely natural in everyday Italian. It is appropriate in speech, messages, and ordinary writing.

It is not slang, and it is not especially formal either.
You could say it to a friend, a coworker, or write something similar in a simple email.

Yes. Italians often simplify sentences when the context is clear.

For example:

• Ho un forte raffreddore, quindi resto in casa.
• Ho il raffreddore, quindi oggi resto a casa.
• Ho un forte raffreddore, oggi resto in casa.

All of these are natural. The original sentence is already very normal and clear.