From the second set of examples, only the second one is correct. Unless you're using
proper nouns ("nombres propios"), you need to use articles to construct your sentences.
La manzana es una fruta.
Manzana is not a proper noun, so it needs the article. If an apple were to be assigned a name, say, its name is Ana (yeah, as in "Ana la Manzana"), then you could get away without articles:
Ana es una fruta.
Now, about the first set of examples...
They are both correct. The difference with the examples in the second set resides in the order of the sentence. For example, both of these are correct:
Llevemos manzanas (Let's bring apples).
Llevemos las manzanas (Let's bring the apples).
They express similar but distinct ideas; the first one states that we should bring apples, but it doesn't specify which apples (maybe they still haven't bought any?). The second assumes both the speaker and the listener know of which apples they are referring to. In fact, if the listener doesn't know of which apples the speaking is talking about, the most natural response would be "¿Cuáles manzanas?" ("Which apples?").
So, with that in mind, let's accept the following examples, which might make it clearer:
Llevemos manzanas para comer durante el viaje (Let's bring apples to eat during the trip).
Llevemos las manzanas que nos dio Roberto para comer durante el viaje (Let's bring the apples that Roberto gave us to eat during the trip).
Going back to the first set of you examples, the first one, "Ahorremos agua", is about saving water, in a very general context. Which water? Well, all of it.
The second example, "Ahorremos el agua", is about saving water too. Which water? It may be, for example, the water they currently have in a bottle while they're crossing the desert! Or, in a very general context, the water of the world (all of it), which would make it equivalent to the first example.