Selecting object components is something you need to do for many things in Maya, but mainly for modeling.
You can do this by switching to Component mode in the Status Line.
The other modes are Hierarchy mode
and Object mode.
A quick way to toggle between the Object and Component modes is to press F8.
In Component mode, you can select a filter to decide which type of component you can select.
Right-click on that filter and select an option to be even more specific about what you can select.
You can also choose the Component filters from the Select menu,
but make sure you're in the Modeling menu set first.
However, it's much faster to use hotkeys,
or right-click on a selected object and use the marking menu to select the component mode.
For NURBS surfaces and curves, right-click and select Isoparm,
Control Vertex,
Surface Patch,
Surface Point,
Surface UV,
or Hull.
For polygons, press F11
or right-click and select Face to select and manipulate only faces.
Press F9 or right-click and select Vertex to select and manipulate only vertices.
And press F10 or right-click and select Edge to select and manipulate only edges.
You can also select polygon components via the Modeling Toolkit,
which you can open by clicking this icon.
Click the Component icons at the top of this toolkit:
vertices, edges, or faces.
Then use the pick/marquee selection tool or the drag tool to select components.
When you select components with the Marquee tool,
you might end up selecting components on the back of an object when you don't want to.
So, how do you prevent this from happening?
You select "Camera Based Selection - On" here,
which also selects it in the Select tool's Tool Settings.
Click this icon or double-click on the tool's icon to open the Tool Settings.
Notice that this is on by default for the Drag tool.
You can also set this as the default behavior in the Preferences window:
Click Selection and select the "Camera-based selection" option.
Now when you select components, they won't be selected on the back side as well.
The Paint Selection Tool is used only for selecting components.
If you're not already in Component mode, clicking this tool's icon automatically switches to it.
Hold down the B key and drag to adjust the brush radius,
then start painting on the selected objects.
As you paint, you'll notice that the vertices remain selected without needing to hold the Shift key.
This is because the default Paint operation in the Tool Settings is Select.
In this mode, you can press Ctrl while you paint to remove vertices from the selection.
The Paint Selection tool keeps you in Component mode as long as it's active,
so to close it, switch to another tool (such as the Select tool), and then you can return to Object mode.
Soft Select is another tool that lets you select and manipulate components.
It's an option that lives in the Select tools, and in all of the Transform tools.
The reason why you can find it in the Transform tools is that it's often used for sculpting geometry,
done by transforming components.
Select components of any type,
then activate the Soft Select option in the current tool.
It gives you a gradient selection for doing proportional editing.
You can also press the B key to toggle Soft Select when any of these tools are active.
Hold down the B key and drag to adjust the radius of selected components,
then use the current tool to transform them.