SVreX gives some very sound advice as does Alfadriver. With that said I'll be a bit of a counter point just to throw out some other things even though I'm pretty much in agreement with them. I apologize up front for belaboring some of the points.
Buying a front running car is a good idea except for the fact that the car set up might not fit the driving style that works best for you. I'll use a friend's Exocet as an example, the person who originally set it up has a polar opposite driving style. I'm a trail braker and he sets the car up to rotate anytime your off the throttle. So first run ever in the car I go storming up to a fast sweeper (50 mph), as soon as I touched the brake pedal I new the car was going around. The person who set up the car was in the passenger seat. When the run was finished we both looked at each other and pretty much said at the same time the car is set up to roll into the corners. I'm lucky in that I can change my driving style almost instantly to fit the car but I prefer a car set up a particular way.
If you do all your braking in a straight line and drive my Datsun (my road race car) you'll think it understeers, additionally you'll likely be freaked out by the brake bias because as part my usual set up I run a lot rear brake bias.
In the Formula 500, which is my autocross car, it's set up with as much oversteer as possible while still being able to get power down coming of the corners. It too has a ton off rear brake bias. Most of this is brought on the fact that F500s have a solid rear axle like a cart so they push the front end in slow corners (all of autocross). If you are not super comfortable with oversteer you will struggle in the car.
A Miata in a street tire class will teach you a lot but as Alfadriver pointed out the will tolerate over driving. You can chuck them round like a spring car, if your driving is at a level where you don't realize you are over driving it you'll never know. You'll happily pound around scratching your head wondering why the times aren't there. Despite that I would recommend one as you will learn a lot in one and once you have a Miata can be made progressively faster. They are also competitive in several classes.
Now if you want to learn get to some sort of school. Here is the rub with autocross because of the low speeds things happen quickly. As someone who instructs the most common thing I encounter is drivers who snap off the brake pedal, which bounces the nose of the car and unloads the front end, instant understeers. At autocross the car will unload the front end and then regain track all within the space of 25ft / 3 tenths of a second. To put this in perspective; on a 45 second autocross course can you give me fed back about 90 things that happened while you where on course. Probably not. Talked to an experienced and fast autocrossers they can tell give you feedback like on 12th corner stay one foot to the left of the 4th cone, you know the one with the smudge that looks like a tiger paw.
For the reasons above I recommend going to a track day; yes the driving style on a road course is different, basically you can't be as aggressive with your inputs. I've coached some friends at track days who are good autocrossers but can't quite make the leap to absolute front runner. The advantage on a road course is things take a looooooong time to happen. If you unload the front end as I described earlier it's going to 80 feet and a full second for the car to recover. Additionally because the corners are 200 feet long it's very easy to isolate the various phases of a corner. Whereas autocross will let you get away with being rough with the car it's becomes immediately apparent on a road course (Not the this stops road racers from being just as rough with the car). Again things happen slower and there is less to process on a road course.
If you local club has an autocross school then go to it, you will learn a lot and it will be a huge help. My instructing isn't purely altruistic, there is a great benefit to multiple points of view. As a driver I'm always trying to learn more, whether it's set up or mental approach.
Fianlly to circle back to the focus of the topic at hand. FWD Civics are excellent cars, the first time I drive one set up for autocross I remember thinking "Holy cow this things a weapon". I like Miatas more because they fit my driving style better. Regardless of car I do recommend a low power car to start. When you make a mistake in a low powered car it's immediately apparent by the tachometer.
So to recap get a low powered car and go to a school, if there are no schools available ask a experienced fast driver to take pity on you and ride along for a run or two. An experienced driver will be able to tell you what to work on after a single run.