Mi madre calienta la sopa en un cazo pequeño y luego la sirve con un cucharón.

Mi madre means my mother, so the possessive mi is needed to show whose mother you mean.

• Mi madre = my mother
• La madre = the mother
• Madre by itself is only used in certain contexts, like direct address or headlines

In everyday Spanish, family members are often introduced with a possessive like mi, tu, su, etc.

Calienta is the present tense of calentar (to heat / to warm up).

Here it is the third person singular form:

• yo caliento
• tú calientas
• él/ella calienta

Since the subject is mi madre, we use calienta.

In Spanish, the simple present often describes:

• habitual actions
• general facts
• actions in a narrative style

So this sentence can mean something like My mother heats the soup... or My mother is heating the soup..., depending on context.

Spanish often uses the definite article more than English does.

So la sopa is natural even where English might simply say soup.

Compare:

• Come pan = He/She eats bread
• Calienta la sopa = He/She heats the soup

Here la sopa refers to a specific soup, probably the one being prepared or served.

In Spain, un cazo usually means a small saucepan or small cooking pot with a handle.

It is a common kitchen word in Spain. It is not the same as:

• olla = pot
• cacerola = casserole dish / cooking pot
• sartén = frying pan

So en un cazo pequeño suggests the soup is being heated in a small saucepan.

In Spanish, adjectives usually come after the noun.

So:

• un cazo pequeño = a small saucepan
• una casa grande = a big house

That is the normal position for a descriptive adjective.

You can sometimes put the adjective before the noun, but it often changes the tone or emphasis:

• un pequeño cazo can sound a bit more literary or emphatic
• un cazo pequeño is the most neutral and natural choice here

Yes, here luego means then / afterwards / next.

In this sentence:

• y luego = and then

It is very close in meaning to y después.

Both are correct:

• ...y luego la sirve...
• ...y después la sirve...

Luego is very common and natural in this kind of sequence.

The la is a direct object pronoun, and it refers back to la sopa.

So:

• la sopa = the soup
• la sirve = she serves it

Spanish often uses object pronouns to avoid repeating the noun:

• Mi madre calienta la sopa y luego sirve la sopa
  This is grammatical, but repetitive.

• Mi madre calienta la sopa y luego la sirve
  This is much more natural.

Because sopa is feminine singular, the pronoun is la.

In Spanish, object pronouns normally go before a conjugated verb.

So:

• la sirve
• lo compra
• las prepara

That is why we say luego la sirve.

With infinitives, gerunds, and affirmative commands, pronouns can attach to the end instead:

• servirla
• sirviéndola
• sírvela

But with the normal present-tense verb sirve, the pronoun comes before it: la sirve.

Here servir is being used transitively: to serve something.

So:

• sirve la sopa = she serves the soup
• la sirve = she serves it

Se sirve can mean something different depending on context, often something like:

• is served
• serves oneself
• part of an impersonal or passive structure

In this sentence, we simply want she serves the soup, so sirve is correct.

Because con expresses the instrument used to do the action.

• con un cucharón = with a ladle

She is using the ladle as a tool.

En usually means in / on / into, so en un cucharón would sound like the soup is somehow located inside the ladle, not that the ladle is the utensil being used.

So:

• calienta la sopa en un cazo = heats the soup in a saucepan
• la sirve con un cucharón = serves it with a ladle

Cucharón means ladle, while cuchara means spoon.

The ending -ón often gives the idea of something larger, though not always strictly.

So:

• cuchara = spoon
• cucharón = big spoon / ladle

A cucharón is the kind of utensil you use to serve soup.

Cucharón has a written accent because the stress falls on the last syllable: cu-cha-RÓN.

In Spanish, words ending in n, s, or a vowel are normally stressed on the second-to-last syllable unless there is an accent mark.

Without the accent, cucharon would be stressed differently, so the accent is necessary:

• cucharón = correct

In standard Spain Spanish, z and c before e/i are usually pronounced with the th sound of think.

So:

• cazo sounds roughly like KA-tho
• cucharón is roughly koo-cha-RON

A few details:

• c before a in cazo is a hard k
• z in Spain is the th sound
• the stress in cucharón is on the last syllable

So a rough pronunciation of the whole sentence in Spain Spanish would be:

Mi madre calienta la sopa en un cazo pequeño y luego la sirve con un cucharón.
≈ mee MA-dre ka-LYEN-ta la SO-pa en oon KA-tho pe-KE-nyo ee LWE-go la SEER-be kon oon koo-cha-RON

Yes, that is grammatically correct, but it sounds more repetitive because la sopa is mentioned twice.

Spanish usually prefers the pronoun once the object is already clear:

• ...y luego la sirve... = more natural
• ...y luego sirve la sopa... = correct but less elegant here

Using la helps the sentence flow better.

The given word order is the most neutral and natural:

Mi madre calienta la sopa en un cazo pequeño y luego la sirve con un cucharón.

But Spanish word order is somewhat flexible. For example:

• Luego, mi madre la sirve con un cucharón.
• Mi madre luego la sirve con un cucharón.

Even so, the original version sounds very natural because it presents the actions in a clear sequence:

1. she heats the soup
2. then she serves it

So while some changes are possible, the original order is probably the best choice for a learner to model.