HomeDNA has the simplest DNA extraction process; just swab each cheek twice with a cotton swab, and place the swabs in the included envelope. The National Geographic Genographic Project sent a scraper that you use on each cheek for 45 seconds, and then place in a vial with stabilizing liquid. MyHeritage DNA has a similar process. 23andMe and AncestryDNA require that you spit into a tube up to the fill line (harder than it sounds), and ship it back with stabilization liquid. Most of the services said not to eat, drink, or smoke for 30 minutes to an hour prior to testing to get the best possible sample.
There are many places you can upload your raw DNA, and several of them are free. Popular third-party DNA analysis tools include GEDmatch and Promethese. GEDmatch is a free, open database and genealogy site that gives additional DNA relative matching and trait results. This tool has information from users of multiple different testing companies. Promethease compares your raw DNA information against scientific reports that link certain markers to health conditions, though you should take these results with a grain of salt as genetic links do not equal a diagnosis.