A Better Drain Lance - Hammerhead
First Available: Chapter 2
Other Requirements: Obtain the Drain Lance
Location: Hammerhead Station
Rewards: EXP; Drain Lance II & Drain Lance III
You'll obtain the Drain Lance after completing Chapter 1 of the game's main story. Once you've obtained this item Cid will begin offering you quests to upgrade it, which will automatically be accepted the next time you speak with him.
For the first quest you'll need to collect a Metal Scrap which are found in many different places throughout the game. I've listed a few further down the page for anyone interested. As for the second quest to upgrade your Drain Lance, you'll need to collect Coeurl Whiskers. These are much more difficult to get your hands on.
A Better Drain Lance: Metal Scrap
A Better Drain Lance II: Coeurl Whiskers
Coeurls are found in multiple locations across the world; Keycatrich in Duscae, Daurell Caverns and Taelpar Rest Area has a den of Coeurl to the east of town. I'd recommend you avoid the Hunters of Secullam Pass hunt which features Coeurls since they have instant death abilities and a butt load of HP making them extremely difficult.
Use my screenshots below for help locating some Coeurls for you to hunt.