First alarm for a fire is 2 Engines, 1 Aerial, 1 Rescue, 1 Hazmat, 1 Air/Light, 1 FRP, 2 DC's (usually). Add an extra engine and aerial for highrise/highlife hazard, highrise will also get the highrise truck if it hasn't been sent as the air/light).
Working fire doesn't add anything unless command requests. It's more or less just a "place marker" for radio comms/incident tracking.
Not sure what 2nd and beyond alarm assignments are, they are so few and far between and alarm assignments. Most of the time command just requests specific additional rigs that are needed instead of requesting a whole alarm assignment. Since they send so many trucks on first alarms now, it's rare they need to call for additional "alarms."
Fire alarm bells ringing are 2 Engines, 1 Aerial, 1 Rescue, if it's a high rise, add another aerial and the highrise unit and if it's a single residential, it's typically just 1 Engine.