Screwdrivers: Pick up a few flat blade screwdrivers and two phillips head drivers. Get a couple small enough to replace the wall plates around light switches, and a couple that are large enough for your bigger jobs. Cheap screwdrivers and wrenches may be the worst cheap tools there are. Try to resist using your good screwdrivers as pry-bars or chisels, use your cheap set for this.
Hammer:
Measuring Tape: At 25 foot long measuring tape is a must have.
Utility or box knife You will want a utility knife with a retractable blade, a solid housing, and in a bright color. These knifes are nice for cutting and trimming things that are too awkward for a saw. You can also remove the blade and use it as a razor blade to clean stickers or paint off of glass.
12″ flat-jaw, arc-joint plier: Most people know these by the brand name Channelocks. While useful for getting a firm grip on various objects they will scar up whatever you use them on. If you use them on a decorative object (anything that looks nice) they will most likely leave unsightly marks on it.
7″ Slip-Joint Pliers: Easy to use pliers which quickly adjust to various sizes.
10-12″ Crescent Wrench: Also known as an adjustable wrench, these will not leave unsightly marks on what you use them on.
Skill Saw: While not necessarily a starter tool, it does come in handy. A skill saw is extremely useful for getting quick clean cuts. Most are also adjustable so that you can get precision angle cuts also.
Hacksaw: Hacksaws are used for cutting metal and pipe, but can be used for just about anything.
Sheetrock Saw: Also known as a sheetrock knife these saws make repairing holes in the wall a thousand times easier. The sharp point allows you to quickly cut into the sheetrock, while the large teeth cut cleanly and quickly.
Electric Drill: A good drill is a definate must. You will once again want a good quality drill. There are cordless drills that are increasing in popularity, however, I usually prefer my corded drill since I know that it will always be ready.
Drill bits: Pick up a small set which will have all of your smaller bits. Generally you can find a small box set with about a dozen or so bits varying in size from 1/16″ to a 1/4″. Also pick up a few larger bits if you plan on doing any framing or running wire.
Wire Stripper and Cutter: If you are planning on doing any kind of electrical work, then this is a must have. Don’t waste time stripping wires with a pocket knife when this simple tool will do it quickly and easily.
Flashlight: There are two types of flashlights that people prefer. Some like the large durable flashlight, while others prefer a small flashlight that can easily fit in their pocket. I personally prefer a small lightweight flashlight, as I can easily illuminate small places inside of the wall and behind things easily.
Shovel: A shovel will probably be one of your most used tools if you do a lot of yard work. Look for a good quality shovel, oftentimes a cheaper shovel will fall apart on you. If you will be moving a lot of dirt or gravel you will want one with a square blade. If you are digging look for a spade shaped blade. For gardening you may want to get a smaller shovel to make it easier to work between plants
Tool Carrier: Now that you have a collection of tools, you need a way to organize your tools. For the occasional do it yourself home repair person a tool carrier might be the most appropriate. However, if you are doing a roofing project, you may prefer a tool belt. There is a nice article here about how to choose between a tool belt and a tool carrier.