What would be a entry-level position for a major in construction management?

If you intend to be a hands on builder then starting out as laborer is the hands on approach. But besides the hands on know how you have to have a good business understanding for insurance, finance and overhead and profit margins. These can be learned in a classroom or by assisting a builder in his estimating and organizing his projects. Management of cost, labor and project organization (scheduling) are key to being successful.

Hello, I started as Carpenter, so I get experience on construction sites, then I go behind CNC wood manufacturing houses, you can look Hundegger.com maschine. When i have both experience and something from air conditioning, and plumbing, started to lead team of carpenter, now i draw and plan houses, making plans in Dietrich's software, but be careful on roofs i fall down from it. If you will have practis on construction field you'll have more perspective of objects in office.

Project engineer which is assistant to the Project / Construction Manager.

But more-importantly you should determine where your greatest interest and likes are.... Management or hands on construction or design or owners representative. They are all involved in construction but with different talents and duties.