Some things are an absolute necessity to have on hand:
Flat-head screwdriver: You often encounter slotted screws on faceplates, switch covers, terminal connectors, and furniture. You never know when you'll have to tighten something, and a flat-head screwdriver works a lot better than a butter knife.
Phillips screwdriver: Many people can't remember what it's actually called, so they refer to it as the "star-shaped thing." Whatever you call it, it's used in many home appliances, such as computer equipment, toasters, and VCRs. Some remote controls have them holding the battery compartment closed, and some appliances use them for fuse covers. It may become your most widely used tool.
Make sure you have scissors. They're often overlooked, but a pair of good, sharp scissors can come in handy in a variety of situations.
Duct tape: Called the universal tool by some, duct tape can be used for just about anything, and often is. It's also available in other colors, if you don't like glaring silver.
Robertson screwdriver: It's the one shaped like a square, and it's often used to assemble furniture.
Torx screwdriver: Torx screws use a series of small ridges to grasp the screw, and are now being used in many small appliances and computer systems. Nothing else will work, so it doesn't hurt to have this tool on hand, too.
Utility knife or carpet knife: It doesn't have to be used only on carpet; you can also use a utility knife to cut tape, linoleum, wallpaper, or just about anything else. Remember that a utility knife is just a razor blade with a handle, so be careful.
Needle-nose pliers: These work great for extracting screws that have fallen into hard-to-reach areas. In a pinch, they can also be used to extract stripped screws—just grab the screw tightly on either side and unscrew it.
Electrical tape: Not only is it great for capping off bare wires, you can also use it to seal pipes in a pinch, either at home or in the car. Electrical tape doesn't rip very easily, so you're better off cutting it with scissors or a utility knife.
Adjustable spanner: Often called a "crescent wrench," it can be adjusted to fit just about any bolt. It's an indispensable tool for cyclists, since it can be used to tighten just about everything on a bike. It's also great for providing resistance when you're trying to tighten a nut and bolt together—hold the bolt with the spanner, and tighten the nut with another wrench.
Spray lubricant, such as WD-40: It's great for getting rid of squeaks in the wheels on bicycles and barbecue carts, and also helps when you're trying to extract tight screws.
Tape measure: "Two grocery bags and a cup and a half on a lard container" isn't a very precise way to order window blinds.
Now that you know the tools to keep on hand, find a place in your home to store them and then always return them to that spot. That way they'll always be there waiting for you the next time you need them.