Capture The Flag competitions describe challenge-based or adventure-based competitions that involve solving a series of technical challenges. After a team has solved a challenge, the team is presented with a flag, hence the name.
Challenge-Based Capture The Flag competitions are comprised of discrete and individual challenges. Each challenge is typically given a point value that will be awarded to the team that solves it. Challenges can be solved in any order; this allows teams to work on challenges individually and of any difficulty. Newbie teams that are trying to gain experience favor this type of competition because it makes it easy to give up on frustrating challenges to work on other challenges.
Attack-Defend CTF Competitions are multifaceted; teams must ensure security over their own infrastructure while finding flaws and attacking the infrastructure of other teams. Challenges are typically services that run on server, and do not follow any order; again allowing teams to work on challenges individually and of any difficulty. These types of competitions are more focused towards advanced teams who won't leave any part of the competition untouched.
Wargames are always online CTF competitions.
Teams can only gain experience three ways: practice, practice, practice. But also, watching presentations about competitions and reading write-ups.
There are many different competitions held all year around, all around the world. The capture.thefl.ag Google Calendar and Forgotten Security's CTF Wiki are good resources to use. Rankings and more information can be found at CTFtime.
There are many different wargames available. WeChall aggregates stats from many of them. Others are listed below.
Many teams keep an online presence and keep updated blogs with write-ups from many different competitions.