Yes, a handy homeowner can cut down a tree, but it should be done only with the utmost care and only for smaller trees. Consider safety as your number one priority. If you have any doubts at all about your capabilities with the size of the tree, call a professional. That said, if you plan to cut down a tree, there are some basic steps to take into account.
Chainsaws are a necessity for cutting most trees. Read the instructions for use and make sure the chainsaw is properly sharpened and tensioned. It's possible to use a lumber saw if the circumference of the tree is no wider than 4 or 5 inches. When using a lumber saw or chainsaw, use a tie-off rope to reduce tension and to pull the tree away from the blade.
Commit yourself to taking all safety precautions. Wear safety goggles, a hard hat, earplugs and closed, protective shoes. Clear the area around the tree of debris and low-level branches. Make sure to have extra gas for the chainsaw. Pre-plan and clear an escape route in case the tree doesn't fall in the intended direction, especially for larger trees. Survey the area for potential hazards such as wires or obstacles in the fall direction.
Planning for the Fall Direction
For trees less than 6 feet tall, the fall direction is determined by the direction the tree is pushed. For taller trees, attach a rope and apply tension by tying it in the direction you want the tree to fall. Make a wedge cut facing the direction you want the tree to fall. Notice, too, the natural lean of the tree, and take it into account by applying a sufficient amount of tension with the rope.
Before beginning to saw, cut a wedge or hinge facing the direction of the fall. Cut horizontally one-fourth of way into the tree's diameter and then at a 70-degree angle about 3 inches above the first cut. Next, move to the back of the tree and position yourself securely and slightly to the side of the path of the planned cut. Holding your chainsaw level, cut toward the previously cut wedge. The cut doesn't have to be completed in one step. It's a good idea to pause half way through to again contemplate an escape route. Finish the cut and move away from the direction of the tree's fall.
Removing the Stump
To remove the stump, let it rot in place and add some soil and fertilizer to speed the process. Other options include renting a back-hoe or stump grinder, or hiring a professional.