The biggest lesson I learned in quickly improving my beer was to control fermentation temperature. My first batch was pitched at 75 degrees and I followed the kit and had the fermentation temperature as measured on the outside of the primary at 72 degrees. While there are a variety of variables, when I started doing batches at 65-66 max (using us-05 or s-04 dry yeast), they started coming out MUCH better. I did a couple kits and then jumped to all grain - every one is different (not trying to say all grain is "better", my reason is that I like scratch baking and that with group buys I could make beer at less than 1/2 kit price via all grain. I use a modified biab method (found here on HBT) where the mash and sparge are done in a bag, which is pretty easy to do.
As also mentioned, sanitation is key and star san (or iodophor) is your friend. Having no-rinse sanitation is SO MUCH easier than when I brewed 15+ years ago and had to use things like TSP and bleach.
As Revy says, always always always take specific gravity readings, it's the best way to know what is going on with fermentation.
I've found beersmith to be well worth the $21 investment in understanding what is supposed to happen given a set of ingredients. (there are alternatives including free which I can't comment on).
Good luck and have fun. I just completed on Thanksgiving batch #10 of the year (and batch #10 in my new brewing era).