The easiest and quickest way to remove unwanted tree stumps is not necessarily the least expensive. You'll want to balance what budget you have for stump removal with the desire for quick removal.
Many tree removal companies have large stump grinders that chip up the stump, including major roots that are just below ground. This can be done within a couple of hours, but can be expensive.
Small stumps (less than 2 inches in diameter) may be pulled out with a tractor or 4-wheel-drive utility vehicle. You'll usually need to do some chopping of brace roots to free the stump.
Stump rotting compounds are generally ineffective. It does help to drill holes into the stump so that moisture and oxygen can get to a larger surface area to begin the decay process.