There are a few ways to increase the Social skill of your colonists, but unfortunately none are as easy or direct as training other skills, like Mining, Construction, Growing, etc.
Neurotrainer mech serum (formerly Neurotrainers) - the quickest way to raise any skill, Social neurotrainer mech serum can instantly and permanently raise a colonist's Social skill. There's a wildcard factor here, though; not only is the serum not common (until you can craft it yourself) and expensive to buy, but you can't tell how much it will raise the relevant skill level; it could be by 1 level, or even 8 levels!
Socializing with prisoners - Capture prisoners and, instead of harvesting them for body parts, recruiting them into your colony, or releasing them for sport out of kindness, set their option to "Friendly Chat", instead. Then your colonists with the "Warden" work type checked will occasionally go and chat with the prisoner, raising their social level.
Relaxing activities - Create relaxing activity opportunities: providing your colonists with a pool table, a TV lounge, a chess board, a social gathering spot, even a single horseshoe pen will all provide opportunities for them to relax and chat with each other. Unfortunately, you cannot force your colonists to be social, so this is, like the rest of the options, not an easy method to utilize.
Related ways to increase Social skills that don't really apply for one reason or another:
Passive boosts - There are occasions where a colonist will become "inspired" for a certain time period in some skill. During this time, any work type that uses the inspired skill will be done faster and better. This skill is typically boosted either by experience gain or level.
Colonist selection - During pre-game setup, you have the opportunity to select a new colonist or colonists if you don't like the ones you're given by default, or if you want to min-max some of their stats. If you are really getting hurt consistently with your starting colonists' stats, use this method during configuration to make sure you start with colonists that are skilled enough to know a boomrat from a pitfall trap.