Definitely working under experienced people. Apartments, condos, new homes, homes being renovated, property managers.
I think a handyman should have a minimum of 8 to 10 years experience and be able to handle light electrical jobs such as adding a circuit, changing outlets or switches, change or replace windows, doors, cabinets.
In addition, plumbing is a big factor. A really good handyman can replace your old tub/shower with a new one, including the carpentry and plumbing required.
Painting, including sanding/priming, brush, roller, and spraying should be in his bag of tricks. Definitely have his own tools and truck/trailer, and prepare rough material estimates for projects.