The easiest first step into car audio is not a component set, although those are basically the only speakers any enthusiasts consider. What you might want to look into are coaxials. These are speakers where the tweeter and the woofer are combined into one speaker. Usually the tweeter sits on top of the woofer and it's permanently attached. The hook up works like a single hook up, so all you have to do is cut the ends off your factory wiring, strip away some of the outer jacket, and screw them down on the coaxial terminals (or some other attachment type). If they aren't screw down, you may need to splice on a connector, but you can find these at walmart, home depot, etc. in the electrical department.
Here's some pictures of coaxial speakers:
And here is a full 2 way component speaker set (6.5" woofer, 1" tweeter, and the "mandatory" crossover that goes with them):
I put mandatory in quotes because more advanced systems (called "active" systems) don't use those little boxes. The box is called a passive crossover. It's like the traffic cop for high frequencies and low frequencies. It receives the full signal, then sends just the treble to the tweeter and just the bass to the woofer. In one of those pictures above you can see 2 circles above the woofer. This is called a 3 way coaxial. It has a tweeter to handle the treble, a midrange woofer to handle the midrange, and a midbass woofer to handle the bass. It's crossover directs all 3 parts of the full signal to their appropriate place.
Coaxials also usually run better on head unit power than components, so if an amp isn't in your budget, coaxials fit the bill. None of these things are overly difficult to install though, including an amp, a head unit, or components. Depending on how large the current speakers are, you may even be able to buy a replacement "baffle" that will convert the stock locations from say a 6x9" speaker to a 6.5" woofer and 1" tweeter hole. This would make mounting the tweeters a non-issue for you. You also tend to see a lot more coaxial speakers at the big box retailers like Best Buy. They are generally considered entry level.
Here's a picture of a premade adapter baffle that converts a 6x9" hole into a 6.5" woofer and 1" tweeter hole:
Car audio however is ALL about compromises. You won't achieve concert hall sound quality with coaxials running on head unit power. Something else to consider is that most factory sound systems make a lot of compromises in order to get some kind of bass output from very little power. The aftermarket usually doesn't. The result is, the factory speakers will probably offer more bass than the aftermarket will, at least in your price range. Adding an amplifier to them will increase the bass output, but it most likely will still be less output than the factory speakers. This is the trade-off between
quality sound and bass output. Car makers know that bass sells cars, so they throw all their efforts into giving you bass, even if it makes the rest of the system sound like crap.
Probably the most important thing to remember when starting out in car audio is that there is no "best speaker". The science of audio is very complicated and there are tons and tons of variables. It's still a growing field, and not all of the answers are yet known, but we're getting closer. For every cool thing you see in a magazine article or at a car show, there are 15 things (maybe more like 100) going on behind the scenes to achieve a good sounding system. $1000 speakers will not sound good if you don't account for these other variables. I mean they'll sound better than $100 speakers that don't account for those variables either, but....
PM me if you'd like me to help you walk through the system build. I'm not a competition level expert by any stretch, but I'm about to wrap up my BS in electrical engineering and I've been an amateur car audio enthusiast for years. I've also helped a lot of people go from factory to impressing their friends.
