Now that you’ve decided what type of joystick you want, let’s think about the minimum of controls you need to cope with a plane or helicopter.
First, you need no fewer than four control axes: the standard two axes of the control stick (roll and pitch), the engine throttle, and the pedals that control the rudder. If you want to fly with more or less comfort, you shouldn't try to save money by buying a joystick with fewer axes.
There must also be an 8-way hat switch for surround vision and half a dozen easily accessible buttons to enable the most important features when in flight.
The two control stick axes are available by default even in the cheapest of joysticks, and most of them also have a throttle axis (if there is none, you can replace it with a couple of buttons even though this is not going to be as fast and accurate as a dedicated throttle axis). It is the rudder axis which is most often cut off in the cheapest (and also in the most expensive) joysticks.
You may think you can do without the rudder just fine. You can turn your plane by rolling it, and that's in fact the only maneuver the rudder helps with. However, when you get more experienced, you realize that without the rudder it is simply impossible to perform a regular turn at a constant speed and height, to maneuver on the ground when taking off or landing, to make a tiny turn just to catch an enemy into your gun sight. The rudder also makes some air combat tricks possible so that your enemy was misled as to the true course of your plane and had to miss even if he's sitting right on your tail. Yes, this axis can be replaced with a couple of buttons, too, to mask those inconveniences (at least, you will be able to take off and land), but the difference between the smooth analog axis control and the discrete button-based one is obvious. Besides, it’s no good spending two easily accessible buttons for that purpose.
Let’s see how the rudder is implemented in joysticks. The most common implementation is a twist handle (this is the only variant for regular single-piece joysticks). It is quite handy, especially when performing sudden maneuvers involving the rudder, but lacks authenticity and precision. A twist rudder makes it harder to use the control stick proper as you can twist the latter accidentally without really needing that.
Some HOTAS systems (like the old Saitek X45 and the above-mentioned Thrustmaster T.Flight Hotas X and SpeedLink Black Widow) come with pedals implemented as a swinging bar on the throttle. This rudder implementation does not interfere with the control stick and is somewhat more precise, but may be more problematic in terms of ergonomics. The bar may turn to be not shaped exactly for your hand. By the way, I guess purchasing a low-end HOTAS system, comparable to single-piece joysticks in functionality, may only be reasonable if you want the swinging bar instead of the twist implementation of the rudder.
Then, the most advanced and authentic implementation of the rudder consists of flight pedals proper. For some top-class joysticks these are the only means of having a rudder since they lack the two other implementations described above. Each serial device of this kind available (a couple of Saitek products, an old model by CH Products which is still being produced, and the pedals included with the Logitech Flight System G940) has three control axes: one is the rudder proper and the other two are separate wheel brakes. Hardcore gamers often buy pedal kits produced in small volumes by obscure firms. They have high-quality mechanics and electronics (something which the serial models lack) but the most affordable of them (which are still more expensive than any mass-produced model) lack wheel brakes.
You may be wondering if you can use the pedals from your racing kits for flight sims. Well, the flight pedals work differently. They are connected mechanically. If you push one forward, the other moves backward. And even if you don’t care about authenticity, you will only be able to use car pedals in which the accelerator and brake are assigned to one axis (this is easy to check out: the pedals are implemented using the single-axis method if there is no reaction when you press them both simultaneously).
What I’ve described above is the required minimum, but each extra axis or button above the baseline is going to increase your comfort. Additional axes may be implemented for individual engine throttle control, smooth regulation of the twist, for controlling the position of the wing flaps and trimmers, etc. And with more buttons on the joystick, you’ll be able to use your keyboard less often.