A missão começou hoje.

A is the definite article meaning the.

It is feminine singular because missão is a feminine singular noun. In Portuguese, articles usually agree with the noun:

• o = masculine singular
• a = feminine singular
• os = masculine plural
• as = feminine plural

So a missão = the mission.

Yes, missão is a feminine noun, so it takes a:

• a missão
• uma missão

This is something you simply have to learn with the noun. The ending -ão does not always tell you the gender, because nouns ending in -ão can be masculine or feminine.

Começou is the pretérito perfeito of começar.

In this sentence, it means started or began and refers to a completed action in the past:

• A missão começou hoje. = The mission started today.

It is the 3rd person singular form, used for ele/ela/você and also for singular nouns like a missão.

Because a missão is the subject of the sentence, and começar here is being used normally as an intransitive verb:

• A missão começou. = The mission started.

You do not need -se here.

A form like começou-se would belong to a different structure and would not be the normal way to express this meaning.

The ç is called a cedilla.

It shows that the c is pronounced like s before a, o, or u.

So:

• ca, co, cu normally sound like k
• ça, ço, çu sound like s

That is why começou is pronounced with an s sound in that part.

This happens because the infinitive is começar, and when the verb changes form, Portuguese keeps the soft s sound by writing ç.

In European Portuguese, missão is pronounced approximately mee-SOW̃, with a nasal ending.

A few key points:

• stress is on the last syllable: -são
• ss is a clear s sound
• ão is nasal, which is difficult for English speakers at first

A rough guide is:

• mi-ssão
• IPA: /miˈsɐ̃w̃/

The final sound is not exactly like English ow; it is nasal.

In European Portuguese, hoje is approximately OH-zhuh.

Important details:

• j sounds like the s in measure
• the final e is usually a reduced vowel in European Portuguese, not a full ee sound

A rough IPA transcription is:

• /ˈoʒɨ/

This is one of the words where European Portuguese sounds quite different from Brazilian Portuguese.

Putting hoje at the end is a very normal and neutral word order:

• A missão começou hoje.

Portuguese often places time expressions like hoje, ontem, amanhã at the end, but they can also move:

• Hoje, a missão começou.
• A missão começou hoje.

Both are correct. The version with hoje at the end is especially natural as a plain statement.

Yes. That is also correct.

Compare:

• A missão começou hoje.
• Hoje, a missão começou.

Both mean the same thing. The difference is mainly focus:

• A missão começou hoje. = neutral statement
• Hoje, a missão começou. = slightly more emphasis on today

Usually, in normal Portuguese, no. You would normally say:

• A missão começou hoje.

Leaving out the article sounds unnatural in ordinary speech. You might see article omission in:

• headlines
• notes
• very special styles
• some institutional or military-style language

But for standard everyday Portuguese, keep a.

The infinitive is começar = to start / to begin.

It is basically a regular -ar verb, but with a spelling change in some forms to keep the pronunciation:

• começar
• começou
• comecei

The spelling changes, but the pattern is still very regular. For example:

• eu comecei
• tu começaste
• ele/ela começou
• nós começámos
• vocês começaram

So it is a very useful and fairly straightforward verb to learn.

They are different past tenses.

• começou = started / began
  A completed event
• começava = was starting / used to start / would start
  More ongoing, habitual, descriptive, or backgrounded

So:

• A missão começou hoje. = the mission started today
• A missão começava hoje. = the mission was starting today / used to start today, depending on context

In your sentence, começou is the natural choice because it refers to the start as a completed event.