I have been extract brewing for about two months and have 5 brews under my belt, two of which are bottled and turned out good. My brews have been most LME, all have had steeped grains, and some DME. All have been kits. My brews turn out darker than I think they should but I guess this is from the extract.
How can I move to the next level to make better extract beer?
(Can't start all-grain, we have a 3wk old baby and the wife is tolerating a few hours a weekend, maybe next year)
Neglecting the LME/DME debate, I'll throw in my $.02 worth.
First, I think you need to know what you like or don't like in beer. That means trying some different beers and figuring out what you really want. Luckily, the research is fun.
Knowing that you're not ready to switch to a more time-consuming style of brewing, I'd suggest the following things to help you make better beer.
First, like Yooper said, Try switching from a kit beer to a recipe. It's not that kit beers suck, bur rather that they tend to sit on the shelf a while. Fresh extract is always better than not-so-fresh, moreso with the liquid extracts. You want to find a shop that has bulk extract, and goes through it fairly quickly. Freshness matters.
Second, I found that the greatest improvement in my extract beers came from two things, in this order: (1) Yeast Management, and (2) Temperature control. Neither requires a huge investment in equipment.
Yeast management is simply using only very high-quality dry yeast (the stuff under the lid on that John Bull kit is NOT what I'm talking about), or good liquid yeast cultures
with an appropriate starter, and consistently aerating your wort well when it's cool just prior to pitching yeast. If you ferment in a bucket, a sanitized wire whip is great for aerating your beer. If you're using a carboy, shake it really well for a minute or two just before pitching the yeast.
As for temperature control: Cooler is better (generally) until you get below 60F. Have a cold basement? Perfect. Put the fermenter on the floor. No basement? Try to find a way to keep your fermenter under 65F. (A big rubbermaid container partially full with water and a T-shirt over the fermenter will get you a few degrees of cooling just from evaporation). Your yeast will reward you with better flavor.
Lastly, remember to have patience. Although the 1-2-3 rule is a good place to start, I found that most of my beer was even better at 8 or 12 weeks than it was at 6. Enjoy the hobby, brew what you like, and take small steps. The transition from "drinkable" to "good" beer is easy to make.
