Learn about the different ways of making a selection - some are obvious, such as the Lasso or Marquee tool, but your software may have other less-obvious methods, such as Color Range in Photoshop, which selects a band of colours according to the sample you choose.
Use feathered selections unless you are confident you don't need to blur the boundaries.
A moderate width of around 10 pixels is a good start for most images.
But adjust the feathering to suit the task - for a vignetted effect, use very wide feathering, but if you are trying to separate an object from its background, then very little feathering produces the cleanest results.
Set the feathering before making a selection.
Note that the feathering of a selection tends to smooth out the outline: sudden changes in the direction of the boundary are rounded off.
For example, a wide feathering setting to a rectangular selection will give it radiused corners.
The selected area is marked by a graphic device called "marching ants" - a broken line that looks like a column of ants trekking across your monitor screen: it can be very distracting.
On many applications you can turn this off or hide it without losing the selection: it is worth learning how to hide the "ants".
In most software, after you have made an initial selection you can add to or subtract from it by using the selection tool and holding down an appropriate key.
Learning the method provided by your software will save you the time and effort of having to start the selection process all over again every time you want to make a change.
Examine your selection at high magnification for any fragments left with unnatural-looking edges.
Clean these up using an Eraser tool or Blur tool.
Making selections is easier to control with a graphics tablet rather than using a mouse.