Wij proberen elke maand geld te besparen op energie.

In Dutch, many verbs of attempt or beginning—like proberen, stoppen, beginnen, weigeren, etc.—are followed by te + infinitive. So you say proberen te besparen, not proberen besparen. The pattern is:
• conjugated verb (proberen)
• te
• infinitive (besparen)

The standard collocation is besparen op + the thing you want to save money on. You cannot say besparen in energie or besparen aan energie when talking about cutting costs.
Note: If you drop op and say energie besparen, that means “to save energy” (i.e. use less electricity/gas), not “to save money on your energy bill.”

When you have a te-infinitive clause with its own object, the object typically comes before te + verb.
Pattern: Subject – conjugated verb – object – te – infinitive.
So you get Wij proberen (subject + verb) geld (object) te besparen (te + infinitive).

Both elke and iedere mean “every” before a singular noun.
• elke maand is very common in speech.
• iedere maand can feel slightly more formal or emphatic.
In practice, you can use either one here with no change in meaning.

Yes. If you put a time expression first, you must invert the subject and verb:
• Original: Wij proberen elke maand geld te besparen op energie.
• Inverted: Elke maand proberen wij geld te besparen op energie.
The verb (proberen) stays in second position, and wij follows it.

Both wij and we mean “we.”
• we is more common in informal, spoken Dutch.
• wij is used for emphasis, formality, or contrast (“Wij wel, maar zij niet”).
In writing or when you want to stress who is doing the action, you might use wij.

It’s grammatically fine, but the meaning shifts:
• energie besparen = “to save energy” (reduce consumption)
• geld besparen op energie = “to save money on your energy (bill)”
Drop geld and op, and you’re talking about using less electricity/gas rather than spending less money on it.