Leave the plugs on top of the soil. They look messy for a little while. But they break down and add nutrients back to the lawn. Core aeration pulls plugs of soil out and lets air, water and fertilizer into the soil. The deeper the plug, the deeper the soil is enriched.
Core aerators are available to rent for a few hours.
These machines pull out two inch cores. The machine has a mind of its own so be prepared for a work out. Lawn maintenance companies offer core aeration service. A Maintenance company may cost more but it saves you from being beaten by a machine. Bluegrass lawns need some food to start the year.