Intro
If you want to understand filters in a more intuitive sense, the process is to first understand the response curves you might expect to see from one-, two-, and three-section cascaded filters. Once you have an idea about the response curves, you will have developed some intuition about how poles work and then you can reverse the process (as one person said to me, "turn things inside out") and learn how to start with a bandwidth or needed rejection, figure out the number of poles you'll need, determine the Q, the center frequency, and pole separation numbers.
In short, you start out by first understanding response curves. Once you get enough of that, you will be in a better position to figure out how to back your way into those from the usual specifications you are more likely to have on hand.
Bandpass Discussion
A one-pole bandpass filter section (also known as a second-order bandpass filter) lets you control only the center frequency \$f_{_0}\$ and the \$Q\$ (or, its inverse, the damping factor \$\zeta\$.)
With two such sections, assuming you treat them as separate sections and not as a single section with two poles, you can control \$f_{_0}\$ and the \$Q\$ of each section, individually. In this case, it's usually better to keep the \$Q\$ the same for both sections and to generate the bandpass shape you want by moving the two center frequencies, one to the left and one to the right of the middle of your bandpass region, in symmetrical fashion (geometric, of course.)
And for three such sections, where you can control \$f_{_0}\$ and the \$Q\$ of all three sections, you'd probably place one section's \$f_{_0}\$ right in the middle and the other two at either end of your bandpass region. Here, you will probably want the \$Q\$ of the centered section to be one-half the \$Q\$ of the two outer sections so that its contribution to the bandwidth is twice that of the two outer sections.
Before I get too deeply into this (and I probably will stop soon, as the point is likely made sooner than later), please note that in the two- and three-section cases, the further we separate the \$f_{_0}\$ from each other, there is an increasing "dip" between each bandpass region. I'm sure this makes intuitive sense to you, yes? As you move the center frequencies of two sections further apart from each other, then you will expect there will be some "voltage sag" in between their center frequencies. If you keep them closer together, then perhaps no sag at all. Or nothing much, anyway.
This is the whole idea of adding more sections. The more you add, the closer the separation is (or the wider the bandwidth), and the dips in between (usually given as flat, or else \$1\:\text{dB}\$, \$2\:\text{dB}\$, or \$3\:\text{dB}\$) will have a lot to do with the design choices you make.
Think of this like "poles" or "sticks" upon which are drapped some sheets of cloth. The closer the spacing of the poles, the less "droop" of the cloth in between them. And if you space them further apart, then there will be more sag of the cloth in between the poles. It's kind of like that.
Summary
I think you should start by just playing with a single pole bandpass filter transfer function. The simplest one you can find:
$$G_s=K\:\left[\frac{2\zeta\,\omega_{_0}\,s}{s^2+2\zeta\,\omega_{_0}\,s+\omega_{_0}^2}\right]$$
Just set \$K=1\$ here to simplify it further. If you want some math to work with, see here for more details.
Once you follow more how filters are shaped, then you will be in a much better position to understand how to start with a shape you want and work backwards to the number of poles you'll require and have a better idea how to cascade them.
This brings up another point. Cascading sections and the spans between them (the dips you allow) relate to insertion loss. You may need to add gain at some point to compensate for the insertion loss of your filter system. For example, the center frequency attenuation will follow this table I just cobbled up:
$$\begin{array}{c|rr}
\text{Insertion Loss} & \\
& \text{2-Pole} & \text{3-Pole}\\
\hline
\text{maximally peaked} & 0\:\text{dB} & 0\:\text{dB}\\
\text{maximally flat} & 7\:\text{dB} & 18\:\text{dB}\\
1\:\text{dB dip} & 11.2\:\text{dB} & 24.2\:\text{dB}\\
2\:\text{dB dip} & 14\:\text{dB} & 28\:\text{dB}\\
3\:\text{dB dip} & 17\:\text{dB} & 31.2\:\text{dB}
\end{array}$$
Anyway, that's about all I want to say for now. It's actually not terribly hard. It's just that most books get overly-algebraic about things and forget to be more graphical and intuitive. But just go with it. Spend some time to understand the impact of moving the center frequencies closer or further apart, or playing with the value of \$Q\$ for each filter section. It won't take long to develop some basic intuitions. Once you have those, you'll find the mathematics a lot easier to interpret and the development of a custom filter design a lot easier to develop on your own.
