Je vais appeler l’assurance demain matin pour avoir plus de renseignements.

Yes, aller normally means to go, but in French it is also used to form the near future:

aller + infinitive

So:

• je vais appeler = I’m going to call
• literally: I go to call, but that is not how it is understood in modern French

This is one of the most common ways to talk about something you plan to do soon.

Both are possible, but they are slightly different in tone.

• je vais appeler = I’m going to call
  More immediate, common in everyday speech, often used for a planned or likely next action.
• j’appellerai = I will call
  Also correct, but a little less conversational in many contexts.

With demain matin, both work:

• Je vais appeler l’assurance demain matin.
• J’appellerai l’assurance demain matin.

The first one often sounds a bit more natural in spoken French.

Because assurance starts with a vowel sound. In French, la becomes l’ before a vowel or mute h. This is called elision.

So:

• la voiture
• l’assurance

This avoids a clash of sounds and makes the sentence flow more naturally.

Here, l’assurance usually means the insurance company / the insurer in a practical sense.

In English, we often say:

• I’m going to call the insurance company
• I’m going to call my insurance

French can also use l’assurance in this way. So in this sentence, it most likely means the company or provider, not the abstract concept of insurance in general.

Because appeler normally takes a direct object in French.

So you say:

• appeler quelqu’un
• appeler l’assurance
• appeler mon ami

Not:

• appeler à quelqu’un
• appeler à l’assurance

This is different from some English patterns where learners may expect a preposition.

Demain matin means tomorrow morning.

French often uses time expressions like this without an article:

• demain = tomorrow
• demain matin = tomorrow morning
• demain soir = tomorrow evening

You do not say demain le matin here.

This is just the normal fixed expression.

No, it can move around.

Your sentence has:

• Je vais appeler l’assurance demain matin...

But you could also say:

• Demain matin, je vais appeler l’assurance...
• Je vais demain matin appeler l’assurance...
  This is grammatically possible, but less natural in everyday speech.

The most natural positions are usually:

• at the beginning for emphasis
• after the object, as in your sentence

Pour means in order to / to, and when the subject is the same in both parts of the sentence, French often uses:

pour + infinitive

So:

• Je vais appeler l’assurance pour avoir plus de renseignements.
• I’m going to call the insurance company to get more information.

The person doing both actions is the same: I will call, and I want to get more information.

Yes, absolutely.

• pour avoir plus de renseignements = very natural, everyday
• pour obtenir plus de renseignements = also correct, slightly more formal or more precise

Both mean something like to get more information.

After expressions of quantity such as plus de, French uses de, not des.

So:

• plus de renseignements = more information
• beaucoup de temps = a lot of time
• moins de problèmes = fewer problems

This is a very common French pattern.

Renseignement means information, details, or information/helpful details.

In French, it is often used in the plural:

• des renseignements = information / details
• plus de renseignements = more information / more details

Even though English often uses information as an uncountable noun, French commonly uses renseignements in the plural in this kind of context.

They are similar, but not always identical.

• renseignements often means practical details, specific information, or information you ask for
• informations is a broader word meaning information in general

In a sentence about calling a company for details, renseignements sounds very natural.

For example:

• Je voudrais des renseignements sur votre assurance.
  = I’d like some information/details about your insurance policy.

Because after je vais, French uses an infinitive.

The structure is:

subject + form of aller + infinitive

So:

• je vais appeler
• tu vas partir
• nous allons vérifier

Only aller is conjugated here. The second verb stays in the infinitive.

Yes, both l sounds are pronounced.

appeler is roughly pronounced like a-pe-lay.

In je vais appeler, the final r of appeler is the normal infinitive ending sound, and the stress pattern is French, not English.

Also, in connected speech:

• Je vais appeler l’assurance

the words link smoothly, but there is no special spelling change here in the sentence as written.

It is neutral and very natural. It works well in everyday spoken or written French.

Nothing in it is especially slangy or especially formal. It sounds like a normal statement someone would say when talking about their plans.

Yes. That is an excellent natural translation.

Depending on context, plus de renseignements could be translated as:

• more information
• more details
• some more information
• further details

All of those fit well here.