<--------Lives 90 miles north of YNP, and this picture is of the Hayden Valley in springtime. The Park is beautiful any time of year, but the summer tourist crush can make things tedious. It is a large park, and you have several options for things to see, from wildlife to geysers, a huge canyon and waterfall, or valleys and the Lake - there are several different aspects.
Yellowstone Lake is indeed over a gigantic caldera. Doomsdayers love to speculate about when it will blow. All I know is that it was the only disaster scenario in our business continuity plan that didn't have recovery options. When it goes, there'll be little for us to recover!
September would be a lovely month to visit. Spring can be muddy, (but late spring has great fly-fishing), summer is crowded, and winter is very cold. Autumn is beautiful! If you are willing to tolerate the cold, winter is actually my favorite time. Take a old-fashioned (heated) snow coach tour through the pristine, crisp air and shush your way through herds of buffalo in a glittering winter wonderland like no other.
Whatever you do - don't get too close to the buffalo while on foot! Those beasties can run up to 30 mph and WILL gore you!
Oh, and don't skinnydip in any hot pools.... it'll blister your pink parts! Those pools are scalding hot -- and sulfuric.
Have fun!