If you are using the latest versions of Photoshop, Illustrator, InDesign, the things you create in one should migrate over to the others without loss. So, you can import the AI image directly into Photoshop with cut/paste. When you paste into Photoshop, you have the option to place as smart object, or raster object (in a few formats). Try moving between Illustrator and Photoshop this way without creating a PNG first and see what kind of results you have.
If you must use your current workflow, check your settings when you make the PNG. Setting "Art Optimized" or "Text Optimized" makes a difference in how the final image appears. Also check the size of the PNG file. Maybe you need to increase the number of pixels for it to look good in Photoshop.
Moving between Illustrator and InDesign can be done without loss of image quality. You can place an AI file or cut/paste the portion of the illustrator file directly into InDesign. It still keeps its vector properties.
If you need to check the InDesign file before printing, export it as a PDF and take a look at that (but make sure your PDF settings are for printing and not "smallest file size")