Pruning trees can be tricky. Practice is the only way to get good. You need a partner across the street to let you know how much to cut if you are new at it. Changing the shape of a tree is easy, but liking the results is another story. Look around your neighborhood for ideas of the result you would like. No one says you have to be original.
If you want to keep a tree from growing too big, cut it back every year. Another thing to remember is that trees create shade. Sometimes you want the shade, but if you want to thin a few branches out to let sun shine on the lower plants, that is acceptable too. Some must be pruned to grow correctly, while others show most beautifully when left to nature.
Tree removal is a whole different story. Cutting down and disposing of mature trees should be left to a professional who is licensed and insured. Power equipment and heights don’t mix well with the amateurs. Once the tree is cut to ground level, usually a stump remains. To complete the job, the stump is removed with a stump grinder, which actually cuts similar to a large circular saw. It is cut in several directions and shreds up the stump and the shavings that remain can be used to fill in flower beds or hauled away depending on preference. If you have a compost, the shavings are a great addition to that. Put organic materials in a compost pile, allow them to decompose and then add to flower beds to promote growth.
A last note on trees. Please respect them. They will often outlive you. They protect you and give back valuable oxygen and nutrients to the planet. They provide homes and sanctity to other creatures. Take care of them and take pleasure in them.