Can't move shape
I have created new project and placed new rectangle in it. I'm trying to move rectangle using direct selection and path selection tools, but it don't moves. When I try to move using Move Tool - it moves whole page together with rectangle, but I need to move only rectangle. Layer is not locked. How to use move tool in Photoshop?
UPD: layers panel:
Ok, so, now that you have added a layers panel I MUST say that this is NOT illustrator, it's nothing like it when it comes to layers and objects editability, in illustrator you can have multiple objects inside a layer and still move them individually.
In photoshop, you can't (see note below) move each object individually once you draw them inside the same layer...they are merged together. In order to move each object individually, like you would do in AI, you need to have each of them in a separate layer.
Note: When I say that you can't, I mean not as easy and "normal" like in illustrator. In your image for example, in order to move the rectangle, you have to select the Rectangular Marquee Tool, select the rectangle with precision from one corner to another, select the Move Tool, click and drag. Though this will also leave an empty space where your rectangle used to be.