Learn Spanish (Latin America): Nosotros vamos a la fiesta hoy.

Vamos is the 'we' form of the highly irregular verb ir (to go). In Spanish, when you talk about going to a destination, you almost always follow any form of ir with the preposition a (to). Together, vamos a gives us 'we are going to'.

Do I have to include the word nosotros (we), or could I just say vamos?

You can definitely just say Vamos a la fiesta hoy. Because the verb form vamos is unique and only ever matches with 'we', the pronoun nosotros is optional. Speakers often include it just for a little extra emphasis or to start the sentence strongly.

The translation says 'We are going', but isn't there a separate '-ing' form in Spanish?

There is, but the simple present tense in Spanish is much more flexible than in English. Nosotros vamos can mean 'We go', 'We do go', or 'We are going'. For things happening around the current time or near-future plans (like going to a party today), native speakers usually just stick to this simple present tense.

How do I pronounce hoy? Do I say the 'h' at the beginning?

No, the letter 'h' is always completely silent in Spanish. You pronounce hoy exactly like the English sound 'oy' (as in the word 'boy').