There's a few basic points to do with clipless shoes. You're right that you have three things, the shoe itself, the pedal, and the cleats, which are the things attached to the bottom of your shoe, which click into the pedal.
Normally when you buy pedals, the box includes one pair of cleats to attach to your shoes, so you know they will fit the pedals. You just have to make sure you buy shoes which are compatible with the pedals/cleats that you have.
There are two main types of shoes, mountain bike, and road shoes (and their respective pedals and cleats).
Mtb shoes have a recessed cleat, and you have a normal rubber tread on the bottom of the shoes, like this
http://www.tredz.co.uk/prodimg/20517_1_Zoom.jpg. So when you're walking, you can walk around like in normal shoes, and the cleat won't hit the floor. When you get on the bike, the cleats can still attach to the pedal, and none of the other parts of the bottom of shoes, touch the pedal.
Road shoes have no sole on the bottom of the shoe. All they have on the bottom is the cleat. There is no tread on the shoe and the cleat is the only thing that makes contact with the floor when you walk. This makes it incredibly hard to walk as there's nothing to grip the floor with, and you kind of look like a penguin. For me it's even hard walking from my room to where I leave my bike, so I have to take slippers with me.
Mtb shoes are preferred by many because it means you can actually get off your bike and walk around a bit, go to a cafe or a pub, go to the toilet, then hop back onto your bike. Whereas in road shoes, they're designed solely for riding your bike, and I rarely do anything else in my road shoes because when I walk with them it damages the cleat and so the cleats need replacing quicker.
I want to stress that A LOT of people use mtb shoes even though they are riding a road bike. It's just having that option to walk in the shoes which makes mtb shoes really popular.
Mtb cleats and pedals are very small, and so pros go for the the bigger road cleats and pedals so there is less hotspotting on the ball of your foot (but this shouldn't really be a factor to consider unless you're riding all the time). Road shoes are lighter because they have no rubber tread on the bottom, and are also very very rigid. I mean you can't bend the sole at all, it always remains flat. This is to maximise efficiency. Again because the sole is so rigid, its hard to walk with. Having never owned mtb shoes, I can't comment on how big a difference you feel when actually cycling with mtb shoes and road shoes, but I imagine it's probably not that big a difference at this level, and other things like fitness, correct bike position, will be more important than simply the shoes you wear.
Heres a comparison of the two types of shoe.
http://drivemybike.files.wordpress.com/2009/05/cleatcompare1.jpgAs you can see, the road shoe has a significantly bigger cleat, and has no tread.
Mtb shoes have a two hole cleat system. They are commonly called SPD. Any shoe that has two holes
http://thumbs1.ebaystatic.com/d/l225/m/mSVQg0ShU_QbR0aPqqX1Gcg.jpg, are mtb shoes.
The cleats tend to look like this.
http://www.evanscycles.com/product_image/image/d9e/d59/8df/26230/product_page/shimano-sh51-spd-cleats.jpg.
Road shoes have a three hole cleat system. They are commonly called SPD-SL.
http://www.brianrourke.co.uk/img/pds2/PROSALE.jpg.
The cleats look like this.
http://3.bp.blogspot.com/-h1IoOC0RV4A/T13mJnKfgKI/AAAAAAAACbo/qCDcwhrgQS0/s1600/Road+Cleat+Shoe.jpgThe first thing you should decide is whether you want mtb shoes or road shoes. Most people choose mtb shoes for their first pair of shoes because of its versatility and ease of use. At a beginners level, having slightly heavier shoes and having smaller pedals is not really going to make that much of a difference, and you'll never know when you will want to walk around in your cycling shoes. They tend to be cheaper as well, and the cleats won't need replacing very often, compared with road shoes. However with that said, my first pair of cycling shoes were road shoes. Initially after the first 6 months, I so wished I had mtb shoes so that I could do other things in them like walking around. I only bought road shoes because I got a good deal on 2nd hand shoes. But in hindsight, I've now got used to them, and if I could do it again, I'd buy road shoes again. Though I only ever get on my bike, ride around for 2 hours, then come back and get off the bike. I think if I was commuting with my bike in say london every day, where you have to constantly stop at traffic lights etc, I would definitely go with mtb shoes.
Once you have decided which type of shoe you want, then you can start picking pedals/cleats. You must always match the brand of pedal, with the brand of cleats. E.g. shimano cleats only work with shimano pedals, look only work with look.
But you don't need to match shoe brands with pedal/cleat brands. E.g. as long as your shoes are two hole compatible, then any two hole pedal/cleats will work with them.
The most popular mtb pedals are the shimano m520 spd pedals, and the sh51 or sh52 or sh56 (multi-release) cleats. I would completely recommend double sided pedals, even for beginners. Many beginners choose to go for a one sided pedal, one side has a slot for the cleats, one side is flat like a normal pedal. The shimano A530 is the most popular one.
http://www.tredz.co.uk/prodimg/15541_1_Large.jpg. People choose this because they think, oh if one side is a normal pedal then it'll be easier to get pedalling if I can't get the cleat in properly. However clicking the cleat into the pedal quickly becomes very easy to do, and then it gets annoying that you can only click in on one side. E.g. if you stop at a traffic light, when you get pedalling again, you need to spin the pedal around until you get to the side where you can click the cleats into.
The most popular style of road pedals are the shimano SPD-SLs, or the look keos. Be careful because the pedals and the cleats both look the same, but they aren't compatible with each other. The look deltas were the older version of look road pedals, but they no longer produce these (though you might find these 2nd hand).
Road pedals are only one sided, but the pedals always have this thick weighty bit at the back of the pedal. When you stop, say at a traffic light, the weighty back part of the pedal will always hang at the bottom. So you will always just step on the pedal and automatically click in.
You can get completely different looking pedals and cleats, like the speedplays or time, but these are usually very expensive and I wouldn't recommend them