I'll start at the beginning. When you say clips, I assume you mean "clipless." Clips refer to the toe clips of yore. They're mostly gone now, except for clips with cheap platform pedals, and some track racers that use toe clips because they spin at a very high cadence and don't want to risk coming out of clipless pedals on a track bike.

Anyways, here's how the shoe-pedal combo works. Shoes, mountain or road, don't come with cleats, but instead they have screw holes in the soles for cleats to screw into. The pedals actually come with the cleat, so they match the pedals.

Looking at shoes, first, the important thing is to find wants comfortable. Road shoes are generally designed for performance, so they will be shaped as to allow a slight bend in the toes, almost as if you are balancing on the balls of your feet. This adds to the power of the pedal stroke. Some mountain shoes are performance oriented, and will have a similar cant to them. Other mountain shoes are designed to have flat soles, and be more comfortable. These shoes are also easier to walk around with. Mountain shoes generally with have some kind of traction element, like a rubber cleat or sole, to make it easier to walk on trails and such.

On the bottom of any shoe you will find the aforemention holes for screwing in cleats. Most road shoes are compatible with many different cleat styles, although some only are set up for traditional pedals. Most mountain shoes, on the other hand, will only handle one type of cleat connection (SPD...I'll get into the cleat connection styles later). If you are road biking, I recommend getting a road shoe. They provide a firm platform for pedalling, with a variety of different cleat types. The downside is that they are difficult to walk in...but hey, who wants to walk? If you decide that walking ability is a priority, then you can get "sport" mountain shoes. They only accept SPD cleats, which allows the cleat to rest in a nook in the sole of the shoe, making walking easier. Some people like that; I don't the sacrifice is worth it. Plus I'm OCP.

As for the cleat connection style, here it is, as promised. The two main types are LOOK/SPD-SL, and SPD. The LOOK-style has three screws, and a wide, triangular platform. It provides a wide surface area, which makes for a very stable pedalling platform. It is also the traditional road style. SPD uses two screws, place horizontally, with narrow spacing, making for a small, compact pedal. Most road shoes accept both. Here is a diagram. SPD-R is not often used, so I'll skip it. I would recommend getting the LOOK/SPD-SL style cleat. They also offer adjustable cleat position, which increases your ability to customize the fit of your bike/pedals.

As for pedals, I have Shimano "105" pedals. They retail for 90 dollars, and they work pretty well. They use the SPD-SL cleats, although an equivalent LOOK pedal will probably work just as well. There are a number of different pedals out there, including SPD and Speedplay designs, but I have no experience with them, so maybe someone can chime in about those. Some people swear by them, but I personally prefer the traditional design.

Looking at shoes, make sure you get one that fits. Like I said, I recommend a road shoe. Specialized makes an excellent lineup of shoes...their "body geometry" fitting corresponds nicely to my feet. While I think that their shoes are generally designed quite well, everybody fits differently, so they might not be your cup of tea. Your shoes should fit like running shoes, or anything normal athletic shoe--snug, but comfortable. I used to have some Specialized Sport Road shoes that I liked. They were very cheap, and they felt good, except that they are too small. Unfortunately Specialized doesn't make size 50, which is what I need. Tear.

Anyways, sorry about the length. Hope that helps.