First soak your hands in some soapy water (or after a bath will work.) Then trim, file, and buff your nails. Always buff your nails in the same way or there is a bigger chance that they will split. Then use a product to moisturize your cuticles - Sally Hansens is great. Then push back and trim your cuticles. Next wash hands very well with a nail brush (or an old toothbrush will work) to remove all oils. Dry them very well. Apply a thin base coat and take it over the tips of your nails. Then apply two thin coats of your favorite color. Make sure you roll and don't shake your color. Allow each coat to dry completely about 5-10 minutes depending on the brand. Make sure your polish is fresh and not thick on the brush. After applying the color finish with a top coat and again take it over the tips of your nails. Wait about five minutes and use Sally Hansens quick dry or baby oil over the tops.
My favorite brands are Sally for base and top coats and OPI for color. They can last a year or even more if you keep the lid on tight and don't leave the brush out for too long at a time. I've found that the Quick Dry type don't last as long, if you are really worried about longevity nailpolish. Just throw it out when it looks thick on the brush - you can try adding nail polish remover but it'll never be the same, it looses its luster. Good luck and have fun!