Transistor stands for "transfer resistor", and in principle that resistance (collector to emitter) can span a very large range, from microamps to amps. It all depends on the base drive.
If you try to control a transistor by explicitly controlling base voltage, as you have realized the effective resistance will change from very high to very low over a rather small span. How to get around this? Don't try to control the base voltage, but rather the base current. Or, if you do try to control base voltage, use feedback to keep the voltage at exactly the right point. Take your circuit, for example. You've left out a most important component - the load. Redraw the circuit as follows:
simulate this circuit – Schematic created using CircuitLab
The voltage across the load will increase with increasing current. I hope you see that the load voltage cannot get to more than about 4.9 volts. But let's assume that it does, say, 5.9 volts. Then the base-emitter junction will be reverse-biased by about 1.7 volts, the transistor will be turned off, and the load voltage will be zero.
So what actually happens is that, for example, as the zener approaches its operating point, so does the corresponding load voltage, but always about 0.7 volts lower. When the zener voltage equals 5.6, the load voltage equals 4.9, and everybody is happy.
This is all a gross simplification, of course, but at your stage of study take it as true. For instance, the transistor has what is called current gain, the ratio of base current to collector current. So in order for the transistor to provide load current R1 must provide enough current to supply both the zener and the transistor. Not only that, but beyond a certain current level the current gain actually decreases with increasing load current. So for very small load resistances (and "small" depends very much on the transistor) a large R1 simply will not provide enough base current to keep the load voltage at 4.9, and the zener will be starved for current and its voltage will drop until a new equilibrium is established. But that is probably a lesson or two in the future.