Player-versus-player combat has been a staple in Minecraft since the game first featured multiplayer capability. As a challenge which tests players against one another in skill, preparation, and tactics, an aspiring combatant should have a firm grasp of Minecraft's controls, as well as a strong grasp of the new combat mechanics introduced in the Combat Update. In particular, two things require mentioning from previous versions: sword attack speed, and shields.
In all previous versions, players could block attacks with their sword, and had the ability to swing their swords as fast as they could click a mouse. There now exists a slight delay in sword-swinging - approximately half a second - which means attack timing is fairly important. This is especially true with shields, which is now the only way to block incoming attacks; a good basic tactic is to block incoming attacks, and immediately swing at the enemy once or twice, knocking them back. Depending on how this is executed, it may be possible to attack someone between their sword swings, creating a sort of rhythm that they cannot break without engaging in some other way.
There are many approaches to this, and any of them can be overcome by a seasoned fighter, so it is a good idea to find what works best for you, and perfect your technique with it.
So, what about taking the fight to other players? In theory, any server with PvP enabled will allow players to fight each other at will. In practice, it can be highly dependent on the individual server's rules, community culture, and a variety of other factors. In addition, the 1.9 update introduced a large number of changes to the basics of combat in all forms, player or otherwise, which should be mastered. If you are ready to dive in head-first, here are a few good communities to try out, each with their own approach to PvP combat:
An interesting mix of PvE and PvP, GodWars has a game mode focused around being a pirate, hunting for treasure, and possibly hunting for other players in the search for more treasure. Team up with your friends and see who's the superior swashbuckler! The community here is a bit small, but very friendly and welcoming to new players.
In addition, the server has a more fantasy-adventure style mode, if pirates don't strike your particular fancy. Both modes are fun, explained well with in-game help tips, and offer a massive variety of activities to dive into, PvP and otherwise.
A much more hardcore experience, MineWind takes conventional PvP and ramps it up considerably. Dubbed "enhanced Vanilla", every plugin and server rule here is designed to hold on to the classic Minecraft feel, with a particular focus on PvP combat itself. Many of the server's features are not immediately apparent, and are geared more for veteran players looking to be thrown in the fire without any hand-holding. Competition is incredibly fierce, and anything worth having is hard-won here.
WARNING: MineWind is a VERY cutthroat environment, where might makes right, and there are no holds barred. Join if you dare!
Boasting a massive array of combat-oriented game modes, including OPFactions, Mega OPFactions, KitPvP, Exclusive PovertyCraft, SG, and UHC (and the promise of more to come), MCWarside is a good place to get your feet wet. The community is very welcoming of newcomers, and there are many different ways to get your fight on, no matter what your preferences.
The layout of the server is visually a bit noisy in places, but packs a ton of features into the space it uses. Something of a middle-ground experience, MCWarside is good for players who are seasoned fighters, but aren't necessarily into extremely competitive environments. Definitely worth a look!
Keep in mind, many servers have not yet had the chance to update to version 1.9, and are still running version 1.8. While it is currently possible to use old-school fighting tactics on many servers, there is no way to know how long that may last in any given community.