Spanish grammar books will tell you to use vayamos for nosotros commands - but no one does. In everyday speech and writing, the shortened forms of vamos and vámonos are almost always used for both affirmative and negative nosotros commands.
¡Vamos! = Let's go!
¡Vámonos! = Let's leave! Let's get out of here!
Let's look at some examples of proper use:
¡Vamos a divertirnos! - Let's go have some fun!
¡Vámonos chicas! - Let's get out of here girls!
¡Ya paso la hora, vámonos soldados!
It's late, let's get out of here soldiers!
In general, Nosotros commands are the equivalent of "Let's + verb" in English, for example, Let's dance!. In Spanish, they may be expressed in two ways, using the "vamos a + infinitive" construction or the subjunctive. Remember that as with all command forms, negatives are based on the subjunctive and object pronouns are attached to the end of affirmative forms.
The Vamos a + infinitive construction:
In the affirmative form the "Vamos a + infinitive" construction has two meanings. For example, "Vamos a bailar" can mean "We are going to dance" (or We will dance), as well as "Let's dance", depending on the context or the intent of the speaker. It is safer to use exclamation points to indicate its use as a nosotros command. There are two items to remember, however:
1) the reflexive form is ¡Vámonos! (Let's leave!; note the omission of the first s)
2) negative forms are based on the subjunctive, vayamos.
¡Vamos a divertirnos!
Let's have some fun!
Let's leave/beat it/get out of here!
No vayamos a almorzar ahora.
Let's not eat lunch now.
!No nos vayamos!
Let's not leave.
The use of the present subjunctive for nosotros commands:
The first-person plural forms of the present subjunctive can be used to express both affirmative and negative nosotros commands, that is, the equivalent of Let's + verb. Remember that:
1) if object pronouns are used, they must be attached to the end of affirmative commands, and this will always require a written accent in the nosotros forms;
2) the first s of the affirmative reflexive ending is lost, that is -mosnos becomes -monos as in ¡Durmámonos! (Let's go to sleep!);
3) when se (representing the pronoun le or les before another object pronoun beginning with the letter l) is attached to the end of an affirmative command, the resulting ss is reduced to s (¡Mandémoselo!, Let's send it to him/her/them! [not ¡Mandémosselo!]
¡Hablemos de otra cosa!
Let's talk about something else!
¡Comamos afuera esta noche!
Let's eat out tonight!
¡Vivamos en paz!
Let's live in peace!
¡No se lo digamos!
Let's not tell (it to) him/her/them!
Let's sit down!
Let's give it to him/her/them!