Damat çok heyecanlı görünüyordu, gelin ise herkese sakin sakin gülümsüyordu.

Here ise marks a contrast between the two parts of the sentence.

So:

• Damat çok heyecanlı görünüyordu = the groom looked very excited
• gelin ise... = whereas / as for / on the other hand, the bride...

It is softer and more literary than simply using ama (but). It helps set the bride in contrast with the groom.

-yordu is the past form of the progressive.

Very roughly:

• -yor = ongoing action/state
• -du = past

So:

• görünüyordu = was looking / seemed
• gülümsüyordu = was smiling

In a sentence like this, -yordu gives a background description, like painting a scene in the past.

It is related, but here the verb is görünmek, not görmek.

In this sentence, görünmek means:

• to appear
• to look
• to seem

So heyecanlı görünüyordu means he looked excited / he seemed excited.

If you wanted was seen, Turkish would more naturally use something like görüldü.

There is an important difference:

• heyecanlıydı = he was excited
• heyecanlı görünüyordu = he looked excited

The first states a fact about his actual emotional state.
The second describes only how he appeared to others.

So this sentence is focusing on outward appearance.

Because gülümsemek usually takes the person in the dative case:

• birine gülümsemek = to smile at/to someone

So:

• herkes = everyone
• herkese = to everyone

This is one of those cases where English uses a preposition (at/to) but Turkish uses a case ending.

This kind of repetition is very common in Turkish. It often makes the expression sound more vivid, natural, or gently emphatic.

So sakin sakin means something like:

• calmly
• in a composed way
• quietly and unhurriedly

It gives a nice descriptive feel. It is more idiomatic here than a plain single sakin.

The dictionary form is gülümsemek (to smile).

Formation:

• remove -mek → gülümse-
• add the progressive → gülümsüyor
• add past → gülümsüyordu

A useful thing to notice: when a verb stem ends in a vowel, that final vowel often drops before -yor.

So:

• gülümse-
• gülümsüyor
• gülümsüyordu

Turkish usually puts the verb at the end of the clause.

So the structure is roughly:

• Damat = subject
• çok heyecanlı = description
• görünüyordu = verb

and then:

• gelin ise = contrasting subject
• herkese = to everyone
• sakin sakin = calmly
• gülümsüyordu = verb

English word order is much more fixed, but Turkish can move parts around more freely for emphasis, as long as the sentence still sounds natural.

Turkish often leaves out subject pronouns when the subject is already clear.

Here, the nouns damat and gelin already tell you who is doing the action, so there is no need to add o (he/she).

Also, Turkish has no direct equivalent of English the or a/an. Whether something is definite is usually understood from context.

So damat can mean the groom, and gelin can mean the bride, depending on context.

Because gülümsemek specifically means to smile, while gülmek more often means to laugh.

So:

• gülmek = laugh
• gülümsemek = smile

In this sentence, gülümsüyordu fits the calm, composed atmosphere much better, especially with sakin sakin.

Here çok modifies heyecanlı, not the verb directly.

So:

• heyecanlı = excited
• çok heyecanlı = very excited

Then the whole phrase goes with görünüyordu:

• çok heyecanlı görünüyordu = looked very excited

So the sense is not he looked a lot, but he looked very excited.