Si tu caresses doucement le chiot, il se calme tout de suite.

Because this sentence expresses a general result or habitual truth:

• Si + present, present often means whenever/if this happens, this is the result.
• It is similar to English sentences like:
  • If you heat ice, it melts.
  • If you pet the puppy gently, it calms down right away.

So here, the idea is not mainly about one future event, but about what normally happens.

Yes. That version is also grammatical, but it means something slightly different.

• Si tu caresses doucement le chiot, il se calme tout de suite.
  = a general truth / usual result

• Si tu caresses doucement le chiot, il se calmera tout de suite.
  = a more specific future result: If you pet the puppy gently, it will calm down right away

A very important rule in French is:

• after si for a real condition, you do not use the future directly in the si-clause
• so French says:
  • Si tu viens, je partirai
  • not Si tu viendras...

Tu is the informal singular word for you.

It is used when talking to:

• one friend
• a child
• a family member
• someone you know well

So Si tu caresses doucement le chiot... sounds natural if you are speaking casually to one person.

If you wanted to be formal or speak to more than one person, you could use vous:

• Si vous caressez doucement le chiot, il se calme tout de suite.

Because the verb is caresser and the subject is tu.

Present tense of caresser:

• je caresse
• tu caresses
• il / elle caresse
• nous caressons
• vous caressez
• ils / elles caressent

So tu caresses is simply the correct tu form.

A useful thing to notice: in pronunciation, je caresse, tu caresses, il caresse, and ils caressent all sound the same.

French usually needs an article before a noun much more often than English does.

So French naturally says:

• le chiot = the puppy

Even when English might leave out an article in some contexts, French often keeps one.

Here, le chiot refers to the puppy as the object of the action.

Doucement is an adverb. It means gently, softly, or slowly, depending on context.

Here it tells you how you are petting the puppy:

• caresser doucement = to pet gently

So it modifies the verb caresses.

Its position is very natural in French:

• tu caresses doucement le chiot

French often places adverbs like this after the verb.

Because se calmer means to calm down.

• calmer = to calm someone/something else
• se calmer = to calm oneself / to become calm

So:

• Il calme le bébé. = He calms the baby.
• Il se calme. = He calms down.

In your sentence, the puppy is the one becoming calm, so French uses the reflexive form:

• il se calme

Here, se is the reflexive pronoun.

Because chiot is a masculine noun in French.

So the pronoun that refers back to it is il:

• le chiot ... il

Even if in real life the puppy were female, French grammar usually follows the grammatical gender of the noun being used.

If you used a feminine noun instead, the pronoun would usually match that noun grammatically.

Tout de suite means right away, immediately, or straight away.

So:

• il se calme tout de suite
  = he/it calms down immediately

This is a very common French expression.

Be careful: although the words literally look like all of suite/following, you should learn tout de suite as a fixed expression meaning immediately.

The comma is very natural and helpful, especially in writing, because it separates:

• the if-clause: Si tu caresses doucement le chiot
• the result clause: il se calme tout de suite

So the comma makes the structure clearer.

In informal writing, punctuation may vary, but this comma is standard and good style here.

A simple pronunciation guide is:

Si tu caresses doucement le chiot, il se calme tout de suite.
≈ see tyu ka-RESS doos-mahn luh shee-oh, eel suh kalm too duh sweet

A few helpful notes:

• si sounds like see
• tu has the French u sound, which is not exactly like English too
• caresses sounds roughly like ka-ress
• doucement has a nasal ending: doo-suh-mahn
• chiot is roughly shee-oh
• tout de suite is roughly too duh sweet

Also, many final letters in French are silent, so the spelling and pronunciation do not always match in an English-like way.