Any gaffers wanna help this lowly lvl.15 lancer become something great? I really have no idea what I'm doing besides solo questing.
Firstly, you should be progressing the story quest as much as possible. Many of the game's secondary features (Airship travel between cities, Owning a Chocobo, Hiring a retainer so you can post to the market, etc, etc) are all unlocked from that.Secondly, make sure you're doing your class quests. These will teach you practical skills that work in actual end-game environments, as well as in-game skills for your character to learn. Plus you get some good gear from them. Both of these have a special marker in your log that looks like a flame icon.
Lots of people opt to farm events called FATEs, which appear on the world map at random locations and allow anyone nearby to participate. You'll see them as big blue circle markers on the map should they be currently happening, and they last usually around five to fifteen minutes. Currently, it's the fastest means of XP if you travel out to a populated zone, join a FATE party, and go from one event to the other trying to complete them. But it's also arguably boring after 30 or so minutes. If you wish to level this way, Western La Noscea is in your level range (15-20) and usually has one or two parties at any given time of the date. Just do a /shout, ask for a group, and follow them around.
Lastly, if you need practice with your skills, do the Guildhests in the Duty Finder. These are designed to teach players, and you get bonus XP for doing them the first time. Eventually the story quest will require you to use the Duty finder and go into dungeons, so this is a good way to get prepared beforehand. Sadly, I don't play a lancer (yet...) so I can't give you any real class-specific advice.