Learn Spanish (Latin America): Vamos a viajar, hay que hacer la reserva hoy. - We are going to travel, we have to make the reservation today.

Hay que is an impersonal phrase meaning 'it is necessary to' or 'one must'. However, because the first part of the sentence says Vamos a viajar (We are going to travel), it is obvious from the context that we are the ones who need to do the booking. English translates this more naturally as 'we have to'. You could also use tenemos que hacer to specifically say 'we have to make'.

Do I ever need to change the phrase hay que to match the 'we' in the first half of the sentence?

No! The beauty of hay que is that it never changes based on who needs to do the action. It is always hay + que + an infinitive verb (an unconjugated verb, like hacer).

I know vamos means 'we go'. Why does it translate to 'we are going to' here?

When you combine a conjugated form of the verb ir (to go) with the word a and an infinitive verb (like viajar), it creates a structure that talks about the near future. Because vamos is the 'we' form of ir, vamos a viajar translates directly to 'we are going to travel'.

Could I just use a single verb for 'to reserve' instead of hacer la reserva?