"I like to walk my dog."
Translation:J'aime bien promener mon chien.
It's closer to 'take out my dog' which can mean walking, or might just mean to transport them to places other than home. I know that a similar idiom is used in English 'I'm taking the dog out' for walking the dog. But it may confuse some people already confused enough by sortir.
"Sortir" = to go out, to take out. So if you take out the talk, it would probably be understood that you are facilitating the dog's business. In this sense, the term "take the dog for a walk" could be understood as a euphemism for the same thing. But "sortir le chien" would nevertheless be translated as "take the dog out". If does not mean you are necessarily going for a walk, e.g., to exercise the dog.
Because whether a 'de' or 'à' or nothing is used before the infinitive depends on the verb before the infinitive and aimer does not have a 'de' after, it just is followed by the infinitive. J'aime faire quelque chose. http://www.collinsdictionary.com/dictionary/french-english/aimer
Walking your dog is a walk with a purpose. Balader as I understand it is more of a wander with someone or something, meandering. But there are some google hits for it so you could suggest it as an option. Especially for those situations where it's not that you are walking the dog for your dog's benefit, but for where you are going for a wander around just with your dog too.
Promener is walking something or someone, where it is implied that one is leading the other (a baby for example - je promène mon bébé). Se promener is when people walk about (no one is leading the other....). Je me promène - I am taking a walk or I am going for a walk.
It just is not. Promener could be better translated as strolling and it is the term used for walking pets in French.
You walk yourself (je marche) but you « promène » your dog.
Promener has a connotation of a leisurely walk. You use promener for pets/animals or even babies. You use se promener when you are taking a leisurely walk with someone (je me promène avec mon ami).
You only use promener when you walk a pet.