startTimer() isn't starting anything, it is checking the current time. But you call it only when the mouse is pressed!
Rather use a global boolean variable indicating the timer is running. Set it in mousePressed() (the function, not the variable in draw()). There, you can play your sound, and init the prevMillis variable to the current milliis() value.
When this variable is true, in draw(), call your function renamed to checkTimer(), for example. When time is up, reset the boolean variable to false.

This kind of thing is easier to code with classes.
Otherwise, you need an array of prevMillis and threshold.

With the latter solution, an example:

1. float x = 0.0;
2. void setup()
3. {
4.   size(800, 200);
5.   smooth();
6.   fill(#FFFFFF,0,0,5);
7.  
8.   //background(200);
9.   noStroke();
10. }
11.  
12. void draw()
13. {
14.   for (int i = 0; i < TIMER_NB; i++)
15.   {
16.     if (checkTimer(i))
17.     {
18.       text("Time is up for " + i, 50, 50 + 50 * i);
19.     }
20.   }
21. }
22.  
23. void keyPressed()
24. {
25.   switch (key)
26.   {
27.     case 'a':
28.       startTimer(0);
29.       break;
30.     case 'b':
31.       startTimer(1);
32.       break;
33.     case 'c':
34.       startTimer(2);
35.       break;
36.   }
37. }
38.  
39. int TIMER_NB = 3;
40. int[] prevMillis = new int[TIMER_NB];
41. int[] thresholds = { 1500, 2000, 3500 };
42. void startTimer(int idx)
43. {
44.   prevMillis[idx] = millis();
45. }
46. boolean checkTimer(int idx)
47. {
48.   if (prevMillis[idx] == 0)
49.     return false;
50.   if (millis() - prevMillis[idx] >= thresholds[idx])
51.   {
52.     prevMillis[idx] = 0;
    
53.     return true;
54.   }
55.   return false;
56. }

Ah oh, let me see if I can get this to work - thanks for all your help on this! It's a steep learning curve coming from no experience;-)

A page to bookmark: http://processing.org/reference/

PhiLho is right in that it is easier with classes but if you are very new to classes and objects it can be quite daunting.

The sketch below has a class called Timer that has just 3 methods
getRuntime() - get the time (seconds) since the timer was either created or reset
getElapsedTime() - gets the time (seconds) since this method was last called. Usually placed in the draw method so you can measure the time between frames.
reset() - resets the timer to zero

The first timer is created in setup and the second timer is created after 5 seconds has run on the first timer.

HTH

1. Timer t1 = null, t2 = null;
2. 
   
3. public void setup() {
4.   size(400, 200);
5.   t1 = new Timer();
6.   textSize(12);
7. }
8. 
   
9. public void draw() {
10.   background(255);
11.   fill(0);
12.   noStroke();
13.   text("Timer 1", 20, 20);
14.   text("  Run time (s)        " + t1.getRunTime(), 30, 40);
15.   text("  Elapsed time (s)  " + t1.getElapsedTime(), 30, 60);
16.   // After 5 seconds, and only we we have not
17.   // done so already, create the second timer.
18.   if (t1.getRunTime() > 5 && t2 == null)
19.     t2 = new Timer();
20.   if (t2 != null) {
21.     text("Timer 2", 20, 120);
22.     text("  Run time (s)        " + t2.getRunTime(), 30, 140);
23.     text("  Elapsed time (s)  " + t2.getElapsedTime(), 30, 160);
24.   }
25. }
26. 
    
27. public class Timer {
28.   private int currTime;
29.   private int lastTime;
30.   private int lapTime;
31.   private int startTime;
32. 
    
33.   public Timer() {
34.     reset();
35.   }
36. 
    
37.   /**
38.    * Reset the timer
39.    */
40.   public void reset() {
41.     currTime = lastTime = startTime = millis();
42.     millis();
43.     lapTime = 0;
44.   }
45. 
    
46.   /**
47.    * Get the run time in seconds
48.    */
49.   public float getRunTime() {
50.     return 1.0E-3 * (millis() - startTime);
51.   }
52. 
    
53.   /**
54.    *  Get the elapsed time in seconds
55.    */
56.   public float getElapsedTime() {
57.     currTime = millis();
58.     lapTime = currTime - lastTime;
59.     lastTime = currTime;
60.     return 1.0E-3 * lapTime;
61.   }
62. }