Ask for: ask somebody to give something
Ask without for: ask somebody to tell something

Compare:

Don’t ask me for money. (NOT Don’t ask me money.)
Don’t ask me the answer. (NOT Don’t ask me for the answer.)
Ask his name. (NOT Ask for his name.)
Ask for the menu. (NOT Ask the menu.)

Ask is sometimes used without for when talking about asking for sums of money in connection with buying, selling or renting.

‘How much is the house?’ ‘They are asking $50,000.’ (= They are asking $50,000 for the house.)

Ask with objects

Ask can be followed by either a direct object or an indirect object.

Ask a question. (Here ask is followed by the direct object – a question.)
Ask me. (Here ask is followed by the indirect object –me.)

When ask is followed by two objects, the indirect object usually comes before the direct object.

Ask him his name. (Indirect object – him, direct object – his name)
She asked me a favor.

The direct object can come before the indirect object. In this case we use the preposition of after the direct object.

She has never asked a favor of anybody. (= She has never asked anybody a favor.)

Ask can be followed by an infinitive. Different structures are possible.

Ask + infinitive
Ask + object + infinitive
Ask + for + object + infinitive

I asked to go. (= I asked permission to go.)
I asked him to go. (= I wanted him to go.)