Сегодня я останусь у бабушки.

Останусь is the perfective future form of остаться.

• остаться = to stay, to remain, to end up staying
• останусь = I will stay

Russian often uses:

• perfective present forms to express the future
• imperfective future with буду + infinitive for an ongoing or repeated action

So:

• Я останусь = I’ll stay / I’ll remain
  focuses on the result or decision
• Я буду оставаться = I will be staying
  sounds more process-focused and is much less natural here

And остаюсь is present tense of the imperfective оставаться, meaning I am staying / I stay, not future.

So in this sentence, останусь is the normal, natural choice.

Because the verb is reflexive: остаться.

Many Russian reflexive verbs end in -ся or -сь. When they are conjugated, the reflexive part stays attached:

• остаться → останусь
• остаться → останешься
• остаться → останется

The -сь / -ся here is part of the verb itself, not a separate word.

Also, the base ending for я in this future form is -у, and with the reflexive particle it becomes -усь:

• остан-у-сь → останусь

Бабушки is in the genitive case.

That is because after the preposition у, Russian normally uses the genitive when talking about being at someone’s place.

So:

• бабушка = grandmother
• у бабушки = at grandmother’s place / at grandma’s

This is a very common pattern:

• у мамы = at mom’s
• у друга = at a friend’s
• у врача = at the doctor’s

So бабушки is not plural here. It is singular genitive.

Here у means something like at, especially in the sense of at someone’s home/place.

So:

• у бабушки = at grandma’s place

This is one of the most common uses of у.

Be careful: у can also mean something more like by, near, in the possession of, depending on context. For example:

• Я у окна = I’m by the window
• У меня есть книга = I have a book

But in Сегодня я останусь у бабушки, it clearly means at grandma’s place.

Yes, я could be omitted.

Russian often drops subject pronouns when the verb ending already makes the subject clear.

• Сегодня останусь у бабушки. = Today I’ll stay at grandma’s.
• Сегодня я останусь у бабушки. = same meaning, but я is more explicit

Including я can:

• add emphasis
• make the sentence clearer
• sound slightly more contrastive in some contexts

For example:

• Сегодня я останусь у бабушки, а завтра вернусь домой.
  Today I’ll stay at grandma’s, and tomorrow I’ll return home.

So the pronoun is optional, but perfectly normal.

Yes, Russian word order is flexible.

Сегодня is at the beginning because it sets the time frame right away: Today...

Other word orders are possible:

• Сегодня я останусь у бабушки.
• Я сегодня останусь у бабушки.
• У бабушки я останусь сегодня.

These all mean roughly the same thing, but the emphasis changes.

Typical effects:

• Сегодня я останусь у бабушки.
  neutral; emphasizes today
• Я сегодня останусь у бабушки.
  also natural; a bit more conversational
• У бабушки я останусь сегодня.
  stronger emphasis on at grandma’s

So the original sentence is a very normal, natural word order.

This is a very important distinction.

• у бабушки = at grandma’s
  location
• к бабушке = to grandma’s / to grandma
  direction

Compare:

• Сегодня я останусь у бабушки.
  Today I’ll stay at grandma’s.
• Сегодня я поеду к бабушке.
  Today I’ll go to grandma’s.

So:

• у + genitive = where someone is
• к + dative = movement toward someone

That is why у бабушки is correct here: the sentence is about staying there, not going there.

Often yes, but not always exactly.

In this sentence, остаться matches stay very well:

• Я останусь у бабушки = I’ll stay at grandma’s

But остаться also has the broader meaning remain.

For example:

• Он остался дома. = He stayed home.
• Осталось два дня. = Two days remain.

So depending on context, остаться can mean:

• stay
• remain
• be left

Here the sense is clearly stay.

The stress is:

• оста́нусь
• ба́бушки

So the sentence is pronounced roughly like:

• Сего́дня я оста́нусь у ба́бушки.

A rough English-style approximation:

• sye-VOD-nya ya a-STA-nus' oo BA-boosh-kee

A few notes:

• сегодня has stress on -го́-
• останусь has stress on ста́
• у is a short unstressed oo
• бабушки has stress on the first syllable: ба́-

Yes, you could say у бабушки дома, but it changes the feel slightly.

• Сегодня я останусь у бабушки.
  Today I’ll stay at grandma’s.
• Сегодня я останусь у бабушки дома.
  Today I’ll stay at grandma’s home / at grandma’s house.

Usually у бабушки already implies at her place, so дома is often unnecessary.

In fact, у бабушки by itself is the most natural version in many situations.

Not always. In many contexts, бабушка is better translated as grandma because it sounds warmer and more natural.

So depending on tone:

• бабушка = grandmother
• бабушка = grandma

In a family sentence like this one, grandma is often the most natural English translation.

Russian uses бабушка very commonly in everyday speech, without sounding overly formal.