It is okay for the temperature in the room to fluctuate some, within a 5 degree range or so I'd say, but it might be good to know more. What temperatures are you talking about? 60s, 70s? If you fluctuate too much you may create some off flavors, especially if you are getting up into the mid 70s, but if you are say somewhere in the 60s or low 70s, I think you will be okay.
I am not sure what your fermenting setup is like, but you want to try to keep that sealed up as tight as you can and get yourself an airlock. Leaving it half open is probably okay, and some here would say its not an issue at all, but there is that chance that it may allow wild yeast or other bacteria to form in your beer and cause a problem--again its probably unlikely, but it would be best to keep your the fermenter as air tight as you can and get a airlock that will allow the pressure to escape without allowing it to introduce any bacteria.
As far as future steps, if you don't have a hydrometer, get one. After a week or so, it may look like nothing is happening any longer, but that yeast may still be working away. The only way to know is to get a sample of your beer and test the gravity. Basically you are looking for a consistent gravity reading over the course of a few days. Once its consistent, then you know the yeast has done its work and you have a few choices from there. You can get ready to bottle your beer since its done fermenting, you can let it sit there in the primary for another week or so and let the yeast and other things really settle out and clear up the beer, of if you have a second fermentation vessel, you can transfer it to that and let it clear up further--many people here would say this is an unnecessary step, I have mixed feelings about it but I am beginning to fall in the camp that there isn't a need for it as I skipped that process in my last two batches and they have turned out just as clear as transferring to a secondary--and as a beginning brewer I wouldn't bother with this step, I am just mentioning it to be thorough on your options.
That also reminds me, if you do get samples of your beer while its fermenting, which I just use a turkey baster to grab what I need, always sanitize it before you allow it to touch your beer.
Other than that, just relax, you have done all the hard work, now you just have to wait for the yeast to do its work. I was nervous in my first few batches, but once you get the process down, its such a great hobby. Welcome to the obsession!