The tools necessary for this project will depend on the complexity of the installation. It could be as simple as a couple of screwdrivers, but you will most likely need a wire stripper and electrical tape. Other tools that could be useful, depending on your installation, are a soldering iron, Allen or Torx wrenches, an offset screwdriver set, and a door panel tool.
The first priority is to decide which speakers you want to replace. Some cars may come with anywhere from two to six speakers. Usually you'll want to replace pairs of speakers unless you're adding a separate subwoofer. Your speakers are probably on the back package shelf, in the doors, or on the dash. Just look for speaker grills if the old speakers aren't working. You can usually see the front of the speakers through the grills, but if not, most grills can be removed easily with a screwdriver. Measure and write down the speaker sizes.
Some car speakers are in more inaccessible places, so if you're having trouble getting to the speakers to measure them, then you might want to call a professional. One great resource is Crutchfield, and the primary advantage to using them is that they have online and print-based guides that help you look up your vehicle. For almost any vehicle, their guide will tell you what size your car speakers are and which of their speakers will be easy replacements. Better yet, if you order from Crutchfield, they include an easy to follow installation guide, and if any adaptor brackets or wiring harnesses are needed, they supply those for free, too.
If you don't have the money to replace all your speakers, you can either replace the pair of speakers that sound the worst or replace the biggest set of speakers. Replacing the largest set of speakers will be somewhat more expensive than replacing a smaller set of speakers, but the results are likely to sound far better. For example, say all four of your speakers are blown, distorting badly on bass notes. You measure them and find that you have 4 inch front speakers and 6 x 9 rear speakers. Unless you pay top dollar for really great 4 inch speakers, it isn't likely that you'll hear too great of an improvement by replacing them. However, there are plenty of inexpensive, high-quality 6 x 9 speakers on the market, and replacing that set will yield much better-sounding results.
The next decision is which speakers to buy. The easiest method is to replace a pair of speakers with a pair that is an identical size. However, sometimes the magnet on the back of the speaker may be too much bigger than the old speaker, or perhaps part of the front of the speaker sticks out too far for the old speaker hole. If you're buying speakers from a local store, it's helpful to measure all the speaker and hole dimensions and take the measurements with you to buy the new set.