The .ai to Photoshop part is easy, just select an Illustrator file when you use the Open command in Photoshop. As you open it, a dialog box lets you enter the pixel dimensions and ppi resolution you want — while you’re in there, make sure the color Mode is set to RGB; if it’s set to CMYK some options such as saving as PNG might not be available. Any transparent areas in the Illustrator file are preserved after it’s converted to Photoshop format.
The export format part is more restricting. It is no problem to create a PNG with transparency using Save a Copy, Save for Web, or Export As. However, the JPEG format cannot store transparency, so no application can do that.