The server uses the mcMMO plugin, which lets you level up skills by doing the matching activity. Skills are split into combat and non-combat.
/mcstats shows your current level in every mcMMO skill./mcrank shows your rank in each skill./mctop [skillname] shows the top players overall, or in a specific skill./<skillname> shows more info about that skill.You level a combat skill by attacking a creature with the weapon that matches the skill. The exceptions are Archery, which uses a bow and not a crossbow (that is a separate stat), and Unarmed, which needs you to hold no weapon.
Attacking a mob that came from a mob spawner gives zero XP. Only naturally spawned mobs count.
Right clicking with a weapon can give you a short powerup for extra damage.
Acrobatics is different: you level it by taking damage, usually fall damage, rather than dealing it. It improves your defense and your chance to survive long falls.
Taming lets you use trained wolves to attack mobs for XP. You also gain XP for breeding them.
These are about gathering resources:
Fishing and Excavation can give surprising loot at higher levels, such as diamonds.
Mining has a double-drop chance equal to your level as a fraction of 1000. At level 256 that is a 25.6% chance.
Right clicking with a shovel, pickaxe, or axe can trigger a frenzy for a short time. Frenzy gives faster harvesting, a chance of triple-drops for mining, or chopping a tall tree in a single cut.
Place a block of iron, then right click it with the tool, weapon, or armor you want to repair.
Do not repair enchanted equipment this way. Repair will almost always destroy the enchantment, and at low levels it is guaranteed.
Repair uses whatever main resource made the item. Stone tools use cobblestone. Everything else uses its main material, such as iron or gold.
Using a block of gold, you can destroy a piece of equipment and get some of its resources back. Salvage cannot be trained directly. It is the average of your Repair and Fishing skills.
Smelting also cannot be leveled directly. It is the average of your Repair and Mining skills. It gives a chance to save fuel or get extra resources when turning ore into metal. It does not help with cooking other things such as glass or meat. You earn a little Repair and Mining when you smelt ore, but slowly.