La versión correcta es 'I am used to drinking milk in the mornings.' Pero no estoy seguro de que sea la mejor traducción de la oración española. El contexto en el que se usa es bastante específico. Su.Ki ha dado un ejemplo del uso. Te daré otro ejemplo. Una persona me ofrece una bebida, pero sólo tiene leche un su refrigerador. Would you like a drink? I only have milk or water. Don't worry about it, I'm used to drinking milk in the mornings. Si quieres decir que bebes leche cada mañana, diría 'I usually drink milk in the mornings.'

That sentence isn't right grammatically. You can say either 'I usually drink milk in the mornings'. or 'I'm used to drinking milk in the mornings'. This second sentence, with 'used to' followed by a gerund, means either that it isn't a problem for you to drink milk in the mornings, or that it would be a problem if you didn't! For example, if you went to stay with a family where there was never any milk in the house in the mornings, you wouldn't be very happy. You'd miss your milky breakfasts, and you might ask your hosts if you could have some milk, because this is what you are used to.