Red vs Blue
If you want to experience the excitement of player versus player combat in Eve without dedicating your life to a nullsec alliance, Red vs Blue is where you want to be. Two player owned corporations, the Red Federation and the Blue Republic, locked in a mutual war with each other in highsec. The focus is on affordable PvP in small, cheap ships with little necessary commitment. Just pick a side and join a fleet. There are self-imposed rules in order to keep the fights simple, fast and fun. Occasionally the two sides will have themed fights using specific ships or modules, or team up in a "purple fleet" and go roaming around lowsec looking for fights. This is the ground-level when it comes to PvP and the organization is fun and successful enough that it runs off of donations from players.
Often considered and intended to be a lite edition of nullsec sov wars, faction warfare takes place in nullsec between the game's four races: the Minmatar, Gallente, Amarr, and Caldari. Players run complexes in deep space and fight with another faction to increase their control in various solar systems in order to capture them from the other faction for theirs. Faction Warfare recently got a huge revamp, with two expansions dedicated to developing and balancing it. The barrier for entry is extremely low and is a good way to familiarize yourself with the game's mechanics and get used to flying in more dangerous solar systems.
The barrier of entry for incursions is higher than either Red vs Blue or faction warfare, but it's a strong comunity of players and a great way to enjoy the game as a team activity. Constellations across the galaxy are occasionally invaded by an NPC force known as Sansha's Nation. These NPCs are tough and smart, using what CCP referrs to as "sleeper AI" to mimic the behavior a real human might have in a fight. They switch targets, use warp scramblers and ECM and generaly are much tougher to kill than regular NPC pirates. Fleets of players come in and clear systems out until a control meter hits 0%, and then a final boss spawns for the players to kill. When this is complete then the incursions ends and participating players are rewarded. There are two groups that run incursions in empire space, and they differ in how they tank their ships. High SP characters are generally needed to do incursions, but the requirements other than that are fairly low.
Another activity that isn't for brand new players is wormholes. If nullsec politics or highsec PvE is too dull for you than wormholes are a great way to inject excitement into your game while also making fistfulls of money. Wormholes spawn all over New Eden which lead to systems far outside the known galaxy (k-space) known as wormhole space (w-space). There are no stations or NPC factions in these systems, and gameplay is very different here. All NPC pirates have sleeper AI similar to Incursion NPCs, the local chat window is delayed, so it's harder to know if there is anyone else nearby, and you need to probe out the wormholes to exit and enter the area. There are a lot of corporations and alliances dedicated to exploring these wormholes and exploiting them for money. Occasionally these groups run into each other in w-space and duke it out. Because of w-space's massive profitability, battles in w-space between wormhole corps can involve only a couple dozen players, but result in as much ISK lost as a huge 400 player fight in nullsec. It's a different life, and one that very few players ever experience.
Nullsec politics and fights grab all of the gaming headlines because they are bold in-your-face events frequently involving thousands of players, large personalities, and thousands of dollars worth of in-game assets, but Eve Online is more than just the big battles and masterminded plots. If the game fascinates you and you want to try it but are worried about minimum level of dedication it requires, then know that you don't have to go all-in if you don't want to. Eve is the most complex game on the market today. There are hundreds of game systems available and an uncountable number of playstyles to try out. There's definitely one for you to enjoy.