Congrats on the first batch. +1 on not having ice on top. I did the ice bath for my first few batches, which did the job but was highly inefficient. Once you get your equipment up to speed from the kit, you'll definitely want to invest in a wort chiller and some type of temperature control. Swamp coolers work great, but I have noticed a huge improvement with the use of a temperature controlled fermentation chamber.
The wort chiller and fermentation chamber are two things that will really improve your brewing process. The chiller limits the chance for infection, partially due to less time for exposure. Obviously, you will always want to put sanitation first on every process throughout. My current fermentation chamber is an old 5cf chest freezer I picked up off CL for $40. Spent another $25 or so on a homemade ITC-1000 temperature control box. Probably improved the outcome at least twofold.
Only issue now is that I broke down and started kegging (dangerous-much easier to throw a cup under the tap than having to take out a bottle, pop cap, pour, rinse bottle, recycle bottle for next batch...haha). So the chamber is not my serving platform. Oh modern day problems.
Until I find a cheap fridge or freezer, I'm back to the swamp cooler. Keep it dark, keep it steady (temperature that is) and if using a plastic fermenting bucket, keep the handle out unless you want a stinky, rusted holder.
Congrats again and happy brewing!