What does it mean when you first have low iron and low hemoglobin, then with diet, the iron goes up but the hemoglobin count remains unchanged?

It takes time. Your low iron level being corrected is the first stage, but your bone marrow will take time to catch up.
Not enough iron. This is likely a situation where an iron rich diet has improved blood iron levels but not yet resulted in enough iron stores in the bone marrow to generate increased blood production. Also the increased iron may be just keeping up with iron losses (e.g. Menses). A trial of oral iron tablets is usually the next step and also some thought to sources of ongoing iron losses.