Voy a la manifestación ecológica con mi familia el domingo.

Because manifestación is a feminine noun, and its article is la, not el.
In Spanish, a + el contracts to al, but a + la does not contract.

• a + el parque → al parque
• a + la manifestación → a la manifestación (no contraction)

Most nouns ending in -ción (and -sión) are feminine in Spanish.
So you say:

• la manifestación
• la canción (the song)
• la conversación (the conversation)

There are exceptions to gender rules in Spanish, but -ción is a very reliable pattern for feminine nouns.

Adjectives in Spanish must agree in gender and number with the noun they describe.

• manifestación is feminine and singular → la manifestación ecológica
• If the noun were masculine: el evento ecológico
• If it were plural and feminine: las manifestaciones ecológicas

So ecológica matches manifestación (feminine, singular).

The subject of the sentence is yo (I), even though you mention your family.

• Voy a la manifestación con mi familia. → I am going (accompanied by my family).
• Vamos a la manifestación. → We are going (subject = my family and I).

Adding con mi familia just tells us who is going with you; it doesn’t change the subject of the verb.

In Spanish, you normally do not use a definite article (el, la, los, las) in front of possessive adjectives like mi, tu, su, nuestro, etc.

So you say:

• mi familia (my family), mi casa (my house)
• tu hermano (your brother)
• nuestra ciudad (our city)

La mi familia is incorrect in standard Spanish.

Yes, manifestación ecológica usually refers to an environmental demonstration/protest/march.
Common English equivalents:

• environmental protest
• environmental demonstration
• environmental march / climate march

Other close Spanish options are protesta ecológica or marcha ecológica, but manifestación is a very typical word for a public protest in many Spanish-speaking countries.

El domingo can mean either this coming Sunday (a specific one) or on Sunday in general; the context usually clarifies it. In this sentence, it most naturally means this coming Sunday.

For habits or routines, you normally use the plural:

• Los domingos voy al mercado. = I go to the market on Sundays (every Sunday).

If you say:

• Voy a la manifestación ecológica con mi familia los domingos.

you are saying “I go to the environmental protest with my family on Sundays” as a regular habit (every Sunday or many Sundays).
With el domingo, it sounds like you’re talking about one specific Sunday, most likely the upcoming one.

Yes, Spanish word order is quite flexible for time expressions. All of these are correct:

• El domingo voy a la manifestación ecológica con mi familia.
• Voy el domingo a la manifestación ecológica con mi familia.
• Voy a la manifestación ecológica con mi familia el domingo.

The meaning is basically the same; the differences are mostly in emphasis and style. Putting el domingo at the beginning slightly emphasizes when.

• Voy a la manifestación el domingo.
  Present tense, often used for planned future in Spanish (similar to English I’m going to the protest on Sunday).

• Iré a la manifestación el domingo.
  Simple future, often used for predictions, promises, or a slightly more formal or distant future (I will go to the protest on Sunday).

In everyday spoken Latin American Spanish, people use the present and ir a + infinitive more than the simple future for concrete plans.

You would use ir a + infinitive:

• Voy a ir a la manifestación ecológica con mi familia el domingo.

Here:

• voy = I am going (present of ir)
• a ir = to go (infinitive)
• a la manifestación… = to the protest…

In relaxed speech, many people would simply say Voy a la manifestación… if the future time (el domingo) is clear from context.

With verbs of movement like ir, Spanish uses a to show movement toward a place:

• Voy a la manifestación. = I’m going to the protest.
• Voy al cine. = I’m going to the cinema.

En is used more for being in/at a place or for means of transport:

• Estoy en la manifestación. = I’m at the protest.
• Voy en autobús. = I go by bus.

Approximate pronunciation (Latin American):

• manifestación: mah-nee-fehs-tah-SYON

  • Stress on the last syllable -ción.
  • The c in -ción sounds like s.

• ecológica: eh-koh-LOH-hee-kah

  • Stress on LO.
  • The g before i is like an English h in he.

Also, v in voy is pronounced like a soft b: boy (not like English voy with a strong v).