Photoshop : How To Export Vector Shapes Tutorial | Graphicxtras
Illustrator Path method
Custom shapes can be added as a path via the options at the top of the application or use the shape layer option. Select custom shape tool. Select path option (top bar) and apply path and then file menu and export and paths to Illustrator. You can then use the resulting vector file in many other applications. Once you have opened the saved the artworks, the generated AI file has no color fill or stroke, you can add that in many apps such as Illustrator.
pasteboard / Clipboard
They can also be copied to some applications via the pasteboard / clipboard. Select the custom shape tool. Set to layer option and then draw the current selected preset by dragging from top to bottom. Go to the direct selection tool in the toolbar and select all the of the design (points) and edit menu and copy. Go to Illustrator and paste into that.
CC library To InDesign, After Effects etc
Use the CC libraries (found via the window menu). Select the custom shape tool and draw the current preset and then drag to the CC library. You can then go to Illustrator and drag the artwork to the art board and select the first option (convert layers to objects) to keep the artwork editable. You will have to release the clipping mask etc and remove any bounding boxes (nothing is ever made easy) and then you can edit the artwork such as moving anchor points etc.
SVG format - Straight to the Web
You can also save it as a SVG file. Select the rectangle, ellipse etc tool and set to layer option in top left. Create the design (rectangle, ellipse etc) and select file menu and export and use the 'as' option. Select SVG format. The SVG format can then be imported into AI as well as Dreamweaver.
You can also do this via the save command by selecting a format (blend with the background or use as normal raster layers) such as PSD or PNG or TIFF format. Go to the file menu and save as PSD or PNG etc to save current raster artwork. The artworks can be saved for use in PSP, Adobe InDesign, Affinity Photo, GIMP etc via the save command. There is an app called PresetBreeze from Tumasoft that saves the artworks from the CSH format files and that is a great way to transfer all your designs.
Probably the best all purpose format is PSD, the native Photoshop format. You can save raster and vector data with that format. Just use the save as command. Create a vector design using rectangle, ellipse etc / create some raster design layers and then go to the file menu and save as and make certain that the layers is ON in the save dialog.