Flying on Concorde... the sound is like a normal subsonic flight... perhaps the engines are a bit louder.
You are correct, you do not hear any super-sonic booms or any other unusual sounds as you pass through the speed of sound.
The only way of knowing how fast you are going is by:
1) Looking at the Mach Meter at the front of the cabin.
2) Noticing your approach into Paris after only 3 1/2 hours after leaving JFK
I recall a slight "puff" of acceleration as we passed through Mach 1...but the sound was not very noticeable.
The most noticeable effect of flying supersonic is that the skin of the plane continued to "heat up". Standing near the door... or leaning toward the window, I could feel the heat.
Otherwise, The Concorde had the feeling of a regional jet with domestic first class seats. But the coolest part of the flight is when we were invited to the cockpit! Even post 9/11!