When it comes to modifying a car, there's little that does as much as a proper set of wheels, tires and new suspension. Affecting the ride, handling and aesthetics, this combination of parts is literally where the rubber meets the road, so making a wise choice is critical to your overall satisfaction with your vehicle. For some, focusing on the more visible wheel and tire choice means that decisions about suspension are made without much thought. However, making that choice is much more than "How can I get low for cheap?" Between affordability, ease of install, performance, aesthetics and expected sacrifices, there's a good bit to keep in mind before pulling the trigger on your car's suspension.
What Can You Afford?
Obviously, the first thing to consider is your budget for the project. On a popular car for tuning, a search for lowering your car can yield results from lowering springs for as little as a couple hundred dollars to race-spec or airbag suspensions that cost many thousands. For a street-driven vehicle that has aspirations of being a bit more sporty on the daily drive, a solid compromise is a set of basic coilovers from an established brand. For many cars, that's going to come in right around $1000 or less. For my Miata, I chose a set of Flyin' Miata Classic coilovers, their lowest priced option, which provides the height adjustability I needed, but foregoes the adjustability that one would need for more aggressive driving or racing. Lowering springs may seem like a cheaper option, but when paired with a recommended set of aftermarket shocks, the price can match or even exceed the price of basic coilovers. 
Can You Turn a Wrench?
While the idea of doing all of the work on your own car is an admirable aspiration, reality is that many enthusiasts don't have the time, space, tools, or knowledge to do more than basic maintenance. While the majority of Stage 1 suspension mods are a bolt-in affair, there's always a few tricks to every car that can trip up amateurs. For anyone that's used to being under a car, installation of lowering springs or coilovers can be expected to take a few hours at least.
Personally, having limitations of nearly everything required to do major work to a vehicle, I trusted the job to the professionals at Atlanta Speed Company. Although the choice does add additional costs to the bottom line, a trusted shop will be able to handle the installation, setup and necessary alignment all at once.
How Does It Perform?
Obviously, everyone wants the car to perform its best. And yet, performance can mean different things to different people. Do you want to be the fastest around the track? Do you want to ride super low and maintain usability? Do you want a suspension that will last the life of your car? As infinite as the desires are, your choices are often limited by your car. For a car like the Miata, there are hundreds of options, in nearly every category. For some cars though, there may only be one set of lowering springs from a company that went out of business 10 years ago. To put it simply, choosing the best performing suspension for your car is going to take research. Both forums and Facebook groups will offer dozens of opinions and reviews for your specific application.
Generally speaking, a quality set of springs and shocks will provide the best aggressive driving performance for the value. Additionally, that same option is ideal for maintaining an OEM-level of comfort, which is good for late-model vehicles or daily drivers. As a next step up, coilovers provide a stiffer ride, and/or lower ride heights. More expensive coilovers will add in levels of adjustability for track purposes, while also being made of better, stronger materials for longevity. Obviously, a modern air suspension, while expensive, will offer infinite adjustability, and can offer high-levels of performance for the track as well. Although this Flyin' Miata setup is primarily for street use, I paired it with lightweight wheels and aggressive Nitto NT05 tires that work for both spirited drives and light commuting.
Does It Look Good?
Although there is a certain segment where suspension will always only be a consideration of performance, a large portion of those buying enthusiast level suspensions are doing it out of aesthetic consideration. Unfortunately, buying the cheapest coilovers and adjusting them to their lowest setting is not going to make for a comfortable or practical ride.
Having a set of wheels and tires in mind before finalizing your suspension choice is a smart idea, especially if you are going for an aggressive look, where tire-to-fender gap, or wheel-to-suspension gap will be tight. Using a wheel size calculator such as WillTheyFit.com will provide some important figures for measurement as you plan out your setup. Additionally, most model-specific forums will have a wheel/tire/suspension thread that will allow you to see what's been proven to work, and what hasn't.
What Are You Giving Up?
Before pulling out your stock suspension, you should be aware that an aftermarket suspension will ultimately require some sacrifices. OEM suspensions are designed to be comfortable, quiet, and long-lasting. Although a properly installed quality aftermarket should not be the complete opposite of those qualities, one can expect a harsher ride and some new squeaks and creaks from regular use. Springs and shocks will require the least amount of sacrifice, as those setups retain the majority of the OEM unit, replacing only the spring and (for the most part) consumable shocks or struts. With coilovers usually replacing the entire OEM shock body (depending on application), you are largely now at the mercy of the manufacturer's engineers. As with the previous sections, do your research! Make sure that you are purchasing a quality product, but also that you have installed it correctly. In reading forums posts, a large percentage of problems are usually the result of a rushed or incorrect installation.
With proper research, a realistic budget, and a firm understanding of how to turn to a wrench (or knowing when you can't), an aftermarket suspension can transform your car to a degree that few other price-similar options are able. Take your time, use the wealth of information available online, and zero in on your priorities and you'll be enjoying your car more than ever before.