I think your intuition about Bernoulli is right. Recall Bernoulli's equation:
is the fluid flow speed at a point on the streamline,
is the acceleration due to gravity,
is the elevation of the point (the one that has velocity ) above a reference plane, with the positive direction upwards
is the pressure at the point, and
is the density of the fluid at all points in the fluid.
I just got this equation from the wiki. You should definitely double-check this against a textbook.
So, how could you get a handle on each of these variables? Going down the list:
might be something of an intermediate target variable (a variable for which you wish to solve). If you knew , you could, as you say, find the volume rate of flow, which could be integrated, I think, to find the amount of fluid left in the tank at any particular time. So would be nice to know. Let's treat that as the target variable for now, then. . I think you'd be wise to choose to be at the orifice, the circular hole drilled in the bottom; that is, after all, the point at which you'd like to know . So isn't going to change.
Now the pressure is going to change depending on how much water is above the circular hole, right? Perhaps you remember doing force calculations of various kinds of tanks in Calculus II? That's a fairly standard problem, because it illustrates the Calc II method very nicely. Obviously, as the water drains out, you'll have less and less water above the hole, and the pressure should get less and less. (How do you relate force to pressure? The usual equation is , where is the area over which the force is applied.) So that's what changes with time. The last parameter, will, I think, be constant, since you're dealing with liquid water, which is mostly incompressible. You can look up the density of water in the units you choose to use.
So hopefully, this starts to get more clear: you've got Bernoulli's equation, which can relate velocity to pressure. What about the constant on the RHS? I'd just plug in known values of all variables for one instant in time and find out what the constant is. Make sense?