The Painting tools are in three segments, but there are only three primary tools valuable to photographers (my opinion, of course). The Brush tool, the History Brush tool and the Gradient tool.
The Brush tool (B) simply paints the foreground color on the image. The size of the brush can be changed with the [ and ] keys, and the hardness by adding the shift key. Options include the Blend modes, Opacity, and Flow. The Smoothing option introduces anti-aliasing to a curved brush stroke. Differences are not radical and I typically ignore the setting. Adobe says higher settings will create smoother results at the expense of efficiency. The default has never been a problem for me.
There are some additional smoothing options under the gear icon which are mostly ignorable. The angle setting would only be useful if you changed the shape of the brush to something other than circular. The Smoothing and Airbrush settings in the options bar are the same as those in the brush panel. The brush panel offers more options including an array of brush shapes and a variety of techniques for applying brush strokes. Most of these are something to explore as you get more advanced in Photoshop, but can be ignored for simply applying brush strokes. The primary use the brush tool is in masking or possibly adding color on a blank layer. This is often done in Color Mode to avoid imposing actual pixels over the image content.
The History Brush is a tool that allows you to apply a history state to an image. This is a more advanced technique, but a very valuable one for retouching. A filter can be applied to an image and a snapshot created to save that state. Then the filter can be deleted and the history state used as a source for retouching. This is useful to apply the results of the filter selectively to parts of the image without affecting the entire image. The primary options for the History Brush are opacity, flow and a Blend mode.
The Gradient tool applies a graduated fill onto the image or mask based on the foreground and background colors. I use this tool primarily to affect masks. With that in mind I set the gradient option to foreground to transparent which allows me to build up a transfer of tone onto the mask. Foreground to background simply replaces the tones each time you drag the mouse. Other options include the gradient "shape" which includes a radial, angle, reflected and diamond shape. A gradient is applied at 100% at or before the starting point and 0% at the end point. A shape gradient applied to an empty layer can be used to apply special effects to an image.
There is also an array of gradients available for color overlays and the ability to modify the gradients and save your own. These can be useful in B&W conversions as they are "Gradient Maps" meaning they convert the image colors to other colors based on the location of various tones in the image. Using selected tones at selected points in the tonal range allows you to modify the tones at those points.
Other options include Reverse which obviously switches the foreground and background colors. Dithering helps create smoother transitions in areas that do not follow the pixel grid, so that is essentially an anti-aliasing. Transparency allows you to modify the application of the gradient to less than 100% by modifying the Gradient editor settings. This is rarely used in photography, so letting it alone (checked) is recommended.