Они решили поспорить о плане.

Решили is the past tense plural form of решить (to decide).

• они решили = they decided
• It is plural because они means they
• It is past tense because the action is presented as something they did

In Russian past tense, the verb agrees with the subject in gender/number, not person:

• он решил = he decided
• она решила = she decided
• они решили = they decided

After решить (to decide), Russian commonly uses an infinitive to say what someone decided to do.

So:

• решили поспорить = decided to argue / decided to have an argument / decided to debate

This works like English decided to + verb.

More examples:

• они решили уйти = they decided to leave
• мы решили подождать = we decided to wait

This is a question of verb aspect.

• спорить = imperfective
• поспорить = perfective

In this sentence, поспорить suggests a completed, bounded event: they decided to have an argument / argue for a while / get into a dispute.

By contrast, спорить would focus more on the process or ongoing activity:

• Они любили спорить о политике. = They liked arguing about politics.
• Они решили поспорить о плане. = They decided to have an argument / debate about the plan.

So поспорить fits well after решили when the speaker sees the argument as a single event.

Not always. Поспорить can mean anything from:

• to argue
• to dispute
• to debate
• to have a disagreement

The tone depends on context.

For example:

• about a serious conflict → argue / quarrel
• about ideas or a proposal → debate / argue about

In Они решили поспорить о плане, it could mean they had a disagreement or wanted to debate the plan. The sentence itself does not force a very hostile meaning.

The preposition о here means about, and it normally takes the prepositional case.

So:

• план = nominative singular
• о плане = prepositional singular

This is why план changes to плане.

More examples:

• говорить о работе = to talk about work
• думать о семье = to think about family
• спорить о плане = to argue about the plan

Not here. The normal form is о плане.

Russian sometimes uses:

• о
• об
• обо

The choice depends on pronunciation and the following word.

Use об before certain vowel sounds:

• об этом = about this
• об идее = about the idea

But before плане, you use о, because плане begins with a consonant sound.

Yes, Russian can often omit the subject pronoun if it is clear from context.

• Они решили поспорить о плане. = explicit they
• Решили поспорить о плане. = [they] decided to argue about the plan, with the subject understood from context

Including они can:

• make the sentence clearer
• add emphasis or contrast
• sound more neutral when introducing new information

Russian uses subject pronouns less often than English.

Yes. Russian word order is more flexible than English because the grammar is shown largely by endings.

The neutral order here is:

• Они решили поспорить о плане.

But other orders are possible, depending on emphasis:

• О плане они решили поспорить. = It was about the plan that they decided to argue.
• Поспорить о плане они решили позже. = They decided to argue about the plan later.

The basic meaning stays similar, but the focus changes.

A helpful stress pattern is:

Они́ реши́ли поспори́ть о пла́не.

Approximate pronunciation:

• они = a-NEE
• решили = ree-SHEE-lee
• поспорить = pa-SPOR-itʹ
• о плане = a PLA-nye

A few notes:

• unstressed о often sounds closer to a
• the final -ть in поспорить is soft
• плане has two syllables: пла-не

Russian has more than one way to say about, but they are not always equally natural in every context.

• спорить о чём-то is the standard pattern for argue/dispute about something
• про is more conversational and is used especially with verbs like talk:
  • говорить про фильм
  • рассказывать про поездку

With спорить, о is the normal choice:

• спорить о плане
• спорить о политике
• спорить о деньгах

So о плане is exactly what you would expect here.

In this sentence, it means decided to argue / debate / dispute, not bet.

That is because спорить can sometimes relate to disagreement, and in some contexts Russian also has meanings connected with making a contention or wager. But with о плане (about the plan), the meaning is clearly argue/discuss contentiously about the plan.

If it meant a bet, the structure would usually make that clearer from context.