I always find myself starting to look into someone's eyes as I begin to speak with them as to tell them, "I am engaged in this conversation" and then my eyes wander as I am thinking about things to say. If the other person is talking I periodically look away, naturally of course, in order to allow the ease of conversation and the comfort therein to be maintained and not awkward. Usually, people like it if you kind of ignore them for a bit and then get back to them with some eye contact and a, "Oh yeah?" "Really?" "Sure, sure." etc. It makes them think that you are a better listener than you think you are and are really easy to talk to because you aren't overpowering them, scaring them, and are really approachable and low-maintenance when it comes to discussing stuff.
The important thing to remember is to start the first few sentences of the conversation off while looking the person's eyes then to casually look beyond them, or at something else for a bit, while confirming what they are saying with conversation fillers like "uh-huh" "really" "you don't say" and the like. After a few moments give them another look in the eyes and nod your head then repeat the looking away process. Depending on the length of the conversation, you repeat as many times as needed.
At the end of the conversation, you do just like you did at the beginning. As you deliver your closing sentences look intently in their eyes with a nice smile and nodding, or editing your mannerisms to coincide with what you are saying, i.e. curiosity if getting instructions, or being asked a favor, or interest if being asked to go out or something, etc. In a way, this sends a subconscious message that the other person had a great time talking to you even if you really didn't hear what they said. Usually, the other person leaves content and satisfied.