The expression hacer un viaje is usually translated to take a trip, both have the same meaning but you can't state that "hacer" means "to take" in all contexts.
In this context "hacer" is a synonymn of the verb realizar
According to the D.R.A.E
hacer
- tr. Ejecutar, poner por obra una acción o trabajo.
realizar
- tr. Efectuar, llevar a cabo algo o ejecutar una acción
The action that you execute in this case is "viajar" or "el trabajo" is "a trip".
The verbs tomar and coger in relation with a trip are used to imply the action of making a reservation in a transport. If you say "quiero coger un viaje a Florida", it's because you're planning to make reservations through a web page or a travel agency.
tomar
- tr. Utilizar un medio de transporte. Tomar el autobús.
coger
- tr. Tomar u ocupar un sitio u otra cosa.