Learn Spanish (Latin America): Vamos a organizar la fiesta.

This is the core formula for the near future in Spanish: the verb ir (to go) + a + an infinitive verb (a verb in its base form, like organizar). Just like in English we say 'going to organize', Spanish requires the a to link the verbs together when talking about future plans.

Where is the word for 'we' in this sentence?

In Spanish, you often drop the subject pronoun (like nosotros for 'we') because the verb itself tells you who is acting. Vamos specifically means 'we go' or 'we are going', so the 'we' is already built into the meaning.

Could I say 'vamos a organizamos' since 'vamos' is in the 'we' form?

No, you should only conjugate the first verb! In the ir a + infinitive structure, only the verb ir changes to match the person (here, vamos for 'we'). The action verb always stays in its original, unchanged form (organizar).

Can 'vamos a organizar' also mean 'let's organize'?

Yes! In Spanish, saying vamos a followed by an action can mean both 'we are going to do something' (in the near future) and 'let's do something' as a friendly suggestion. The context of the conversation usually makes it clear which one is meant.