1. Have a weapon.
   
2. Look up your weapon on wiki.mabinogiworld.com and look at the upgrades.
   Note that you have upgrade sequences.
   
   • 0~0 means it can only be used as a first/starter upgrade.
   • 1~4 means it can be used for second, third, fourth or fifth upgrade
   All upgrades give a different effect, choose the effect you want.
   Do you want it to be mana-efficient? Do you value maximum damage or critical more? Are you using it as a chain-casting wand? Do you want it to be able to shoot enemies from far far away? Is the amount of ammo important to you? etc etc.
   There are a lot of different upgrades, take your time to study them.
   
3. Choose the upgrade you want and look at its proficiency level. That's the amount you need for the weapon to be upgraded with that type of upgrade. You gain proficiency by using your equipment. 
4. After having proffed your weapon, go to the NPC who provides that upgrade (can be seen in the upgrade list) and have enough gold with you to do the upgrade (amount of gold is seen in the upgrade list)
   
5. "Modify your weapon"
6. Upgrade your weapon with your chosen upgrade.
   
7. Voilà!

So what you need is: your weapon with enough proficiency and money.

Gem Upgrading

After the maximum amount of upgrades, you can do a gem upgrade. There are usually two different gem upgrades. One that gives more durability and one that gives some stats like max damage or critical etc.

• You need money, the gems for your upgrade (can be found in the upgrade list, don't forget the size of the gems!) and your weapon on 100% proficiency.
  
• Upgrade it just like a normal upgrade

Then your weapon is fully upgraded, and special upgrading awaits you~

A few tips:

• Raise the proficiency on your weapon to 100 first, then upgrade. Saves time looking whether you have enough prof and what not.
  
• Balance doesn't matter much. Don't be too concerned about it when an upgrade decreases the balance.
  
• Take your time thinking about the upgrades, since you can't remove your upgrades! (except for Artisan Upgrade, which can be removed with NX)
• I don't recommend upgrading equipments you know you won't be using for a long time.
• When upgrading clothing or armor: note that upgrades can personalize your equipment. That means you wont be able to stuff it away on a pet or trade/sell it to someone else or even put it in your dressing room. You will have to carry your equipment around in your inventory. It can be put in the Me-tab though.
• The same goes for special upgrading: it will be personalized.