With few exceptions, the last three groups of tools are much less valuable to photographers until a level of experience makes them rise in importance. The first group is the Drawing tools. The second is the Type group (T) which may be useful if you like to overlay type onto an image. This is a very easy to remember shortcut so I do not include the type tool in my essential toolbar. The third group is the Navigation group which includes tools that are much easier to access with shortcuts and they are some you should learn first. They are easy.
The primary tool here is the Pen Tool (P) which allows the creation of vectors to follow the outline of a subject. Vectors are shapes that do not adhere to the array of pixels in an image. The tool is very good once you understand it, but it is not intuitive and the result will be a path. A path needs to be converted into a selection before it can be used in a raster image, so other selection tools get most photographers where they need to be without the Pen tool. Other options in the group are simply variations on the Pen tool for specific purposes. Very useful to graphics people for creating "clipping paths" for offset printing.
Additional tools to this group are the Shape Tools (U) which allow creating simple or custom shapes which are saved as paths. They are again primarily vector images and most useful to graphic artists.
The Type Tools (T) include a horizontal and a vertical type tool and type mask tools. Type tools obviously create type within the box you create by dragging on the image. This can be modified later, and the type can be managed in terms of alignment within the box. The type layer can then be moved to any location on the image. The type itself can be controlled in terms of its color, size, font and style. Type is by default vector based. It remains editable unless the image is flattened or the type layer is rasterized. Once rasterized it cannot be rescaled. The type path tools simply allow you to create type which is used a mask which would allow the type to appear as the content of the image below.
On the right side of the options bar are two options to note. The first is the Type Warp option which allows you to make type follow curved or warped shapes rather than a standard baseline. Second is the Paragraph dialog which includes the Paragraph settings for alignment of the text including justification and spacing. It also has a tab for Character control which extends the simpler options for text style to control over kerning (spacing between letters), leading (spacing between lines) and other typography controls. Be careful here as the settings are "sticky" and will affect type applied in the future as well as the current document.
There are three navigation tools, the Hand Tool (H), the Rotate View Tool (R), and the Zoom Tool (Z). I consider image navigation to be the first thing you should learn to manage with shortcuts rather than the tools themselves.
The Hand Tool (H) is accessible by holding down the Spacebar. This allows you to move the entire image around the screen. Note that when the image is either fitting the screen or smaller the image cannot be moved in the Standard screen mode. This is why the full screen modes are useful as the image can be moved to better see image edges. In Standard screen mode this can only be done when the image exceeds the screen size. The Hand Tool should not be confused with the Move Tool in navigating the image as the Move Tool only affects the active layer or layer group.
The Rotate View Tool (R) allows you to rotate the entire image on screen which may be useful for some retouching techniques. The Rotate tool is designed to allow non-destructive rotation as rotating an image using the transform options are destructive. I don't find it particularly useful. Be careful to avoid confusing the Control [Command] R shortcut for viewing the rulers with the rotate tool. It also depends on the ability of your graphics card to manage the rotation. There is a reset button to return the image to normal.
The Zoom Tool (Z) simply places a magnifier icon on the screen which allows you to increase image size in increments or decrease with the Alt [Option] key added. The Control [Command] key and the number 1 render the image at 100%, and used with the Zero key makes an image fit the screen. That may be at an unusual percentage which you should note at the bottom left of the screen. Control [Command] Alt [Option] and the Zero key also render the image at 100%. Control [Command] and the spacebar allow "scrubby zoom" using the mouse movements left and right to resize the image in non-specific increments.
The Control [Command] and the + and - keys modify the image size in increments of 25%, 33%, 50%, 66%, and 100%. as well as some smaller increments as well. The 25%, 50% and 100% increments are most valuable in being able to accurately judge image quality while other percentages are interpolated and do not well represent the image. Control [Command] plus the Alt [Option] key and the zero key render the image at 100% on screen. Only at 100% is the image represented totally accurately and that should be used to determine the settings for filters applied to the image.