I don't know what the burp method is so I can comment on that. You will never go wrong with an airlock. Put some starsan or vodka in it and you are done.
Like billl said, temp is one of the most important parts of fermentation. Each yeast has its own range. I typically like to stay at the long end of the range...typically I ferment in the low 60s.
When we take about fermentation temp, we aren't talking about the ambient air temp. When I say I ferment in the low 60s, that is the temp of the actual beer. If you have a strong fermentation, the beer could be as high as 10 degrees warmer than the ambient room temp. So be careful of that. If your room temp is 68, you might think you are fermenting in a good temp range but you could be up to 78 degrees which is too high.
There are a ton of ways to keep fermentation temp down. You can get a fridge, you can get a chest freezer with temp control, you can use a wet towel with a fan, you can use a water bath putting in frozen water bottles...the list goes on. Right now I am using a cooler with water in it about half way up the bucket. I put in 2 frozen water bottles when needed. Works perfectly. You might get some temp swings but nothing drastic.