Trading between players is not impossible... It's just a bit weird.

When you equip an item, said item will automatically bind to you. When an item is bound to you, you cannot trade it to another player. You can get a Gremlin known as Vice to unbind an item, but the price is steep and not recommended unless you are sure to get your money's worth. This is why Pulsars, Antiguas, and Sealed Swords may seem horribly overpriced in the Auction House (said items are acquired bound).

For PVP, there is a button. Top right corner of your screen (While in haven) you should see two purple crossed swords. Click it and it will bring up the PVP stuff. There are two game modes, Lockdown and Blast Network. You can get full information about both of these games on the Wiki, so here is a nutshell:

Lockdown is all about captures. There are a series of capture points available (minimum of 3), and holding one slowly gives your team points. It works like many other control point games (For example, Team Fortress 2). In Lockdown, you must bring in your own gear (except shields). Any gear you have, you may use. Lockdown is divided into three Tiers (Same as clockworks), and if you are filed into the highest tier possible (For example, if you have tier 1 armor, but one tier 3 weapon, you will be in tier 3). You shields are replaced with one of three possible replacements. There is the Striker Boost, the Recon Cloak, and the Guardian Shield. Striker is the most common class. It gives bonuses to swords, a jetpack to zoom around the battlefield, HUGE penalties to Bombs, and reduced health (compared to the other classes). The Recon relies on stealth. The class allows players to turn invisible and sneak around the battlefield. If you are hit while invisible, your "shield" will break and you will be unable to cloak until your invisibility regenerates. Recons get bonuses to Guns and bombs, but a slight penalty to swords. The last class is the Guardian, which has the most health and the Guardian Shield. The Guardian Shield provides AOE defence and slowly heals any ally in it. Guardians get bonuses to swords and bombs, but a slight penalty to guns.

Blast Network is a completely different game. You are provided with all the gear you use, so no one has an unfair advantage. The object of the game is to... well... blow up the other players. It's not exactly the easiest to explain, and I've heard of several people meeting some humiliating early deaths when first learning it. I personally played it once, and haven't since.