Werkwoord

Hi Ludmilla!
I don't really know the rules, but thinking about some examples, I seems that you should separate it, when you don't have another verb in the sentence. I hope these examples will help you a little bit, but maybe someone else can give you the exact rules.

Aandoen

I am putting it on. -   Ik doe het aan.
I am going to put it on.  -  Ik ga het aandoen.
I can put it on.  -  Ik kan het aandoen.
I want to put it on.  -  Ik wil het aandoen.
I am allowed to put it on.  -  Ik mag het aandoen.
I have to put it on.  -  Ik moet het aandoen.

Werkwoord

This is something that I find a little hard to explain, but I'll do my best. The form these verbs are written in when they are whole is called "infinitief", which is the standard form of the verb. All verbs have this form, not just verbs that can be separated. Take the verb to walk for example: lopen (this is the standard form: infinitief). In some constructions you need to use this form of the verb, like when you use a "hulpwerkwoord", literally helping verb:

Ik ga lopen - I am going to walk

In this case you would also need the infinitief form of verbs like aandoen, because of the construction you are using. You use the verb as a whole:

Ik ga aandoen - I am going to put on

One exception to this is when the word "te" is used:

Ik vind het prettig om te lopen - I like to walk

In this case the verb aandoen would be separated and te would be put in between. The words the verb are made up of are still kept in the same order though:

Ik pak een T-shirt om aan te doen - I am grabbing a shirt to put on

When you use the present/past simple or imperative of the verb, you often need to separate it. In this case, the order of the words changes as well:

Ik doe een T-shirt aan - I am putting on a shirt

Hij deed een T-shirt aan - He put on a shirt

Doe je jas aan! - put on your coat!

But, again, there are exceptions. Verbs like these have a specific conjugation for subordinate clauses (I believe that's the word in English) that other verbs don't have. Take the sentence "If you put on your coat, you won't get cold." The first part of this sentence would be the subordinate clause. It is translated as "Als je je jas aandoet, krijg je het niet koud." As you can see, the verb is kept as a whole.

I hope this helps a bit, but I mostly advise you not to worry about it too much. I think this is something you will learn with experience. :)