First, start with a basic Claw Hammer. Hammers are for, well, hammering, usually nails into things. The claw part on the opposite side from the head is used for extracting nails. Hammers can also be used to knock things apart, which can be a rather fun way of relieving stress, though that is strictly an "off-label" use! For your starter kit, I suggest a 7 oz. hammer, though as you build your tool kit you'll want to add a heavier hammer and perhaps specialty hammers at some point.
Next, you need some screwdrivers. At a bare minimum for your starter set, I suggest four - two Phillips head (the ones that like kinda like a cross) in sizes #1 and #2, and two flat head screwdrivers (the ones with a straight edge) in sixes 3/16 and 1/4. You can buy screwdrivers in various lengths, but you'll probably start with six-inch ones. As you add to your tool kit over time, you'll probably want to add different lengths, sizes, and styles (yes, there are many more styles than just the most common Phillips- and flat- head screwdrivers). You can also buy screwdrivers with interchangeable bits, whcih can be a space saver.
You will also want a pair of needle-nose pliers (six-inch) and regular pliers (six-inch). Pliers are great for gripping, holding, bending, and twisting things (which can be surprisingly useful activities). The needle-nose pliers typically also come with built-in wire cutters/strippers, which can also be useful. Both regular and meddle-nose pliers come in various styles and sizes, and you'll be adding to your starter kit over time.
An adjustable wrench should also be part of your starter tool kit. Like other tools, they come in various sizes, and you'll probably get a larger one at some point (the big ones can get surprising expensive), but you should be okay starting out with the standard six-inch in length with a 3/4 inch opening.
A retractable tape measure will prove to be an invaluable addition to your starter tool kit, provided you use it, as it wil keep you from making too many mistakes. You can get them in almost any length imaginable, but I recommend a ten- or twelve-foot one for your starter kit. You can get longer ones later, if you need to...
A bubble level, sometimes called a spirit level, will help you make sure that pictures frames, tables, and other items are level. You could use a marble, but a level really is more convenient.
An Allen wrench set, also called a hex-wrench set, are those weird-looking, L-shaped, hexagonal pieces of metal, that are surprising useful for putting together those some-assembly-required pieces of furniture you buy at the big box stores, among other things. They come in both SAE and metric sizes, and you'll want a set of each.
You'll also want to add a good utility knife/box-cutter to your starter kit. Great for cutting things. I also find a good, sturdy pair of scissors quite useful.
A final item to consider for your basic starter tool kit is a set of precision screwdrivers. There screwdrivers are especially deigned to fit very small screws, like those found on eyeglasses & sunglasses, and in watches & various electronics. Like regular screwdrivers, precision screwdrivers come in various lengths, sizes, and styles. For your starter tool kit, just get a basic set.
At this point, your most basic starter tool kit is complete.You'll probably want to add a bag, box, or case of some sort to store your tools in, to keep them together so you can actually find them when you need them.
You'll also find it handy to go ahead and buy a variety of hardware (items like nails, screws, bolts & nuts, picture-hanging wire, etc). You can typically find small kits of various hardware at the big-box stores. Don't forget duct tape!
For a more complete discussion of tools, see I recommend the book by James Wesley, Rawles Tools for Survival.
As you build your tool kit and workshop. don't forget about safety equipment like protective eye-wear, gloves, aprons, fire extinguishers, smoke alarms, and first aid kits! (Seriously, safety is important. See my article Prepper Workshop and DIY Safety.)
The tool kit pictured at the top of this article is the Stanley 65-Piece Homeowner's Tool Kit, which mostly meets and even exceeds my criteria for a basic starter tool kit. In addition to the basic tools I mention in this article, it also has some additional drivers and a small socket set. The only thing missing that should be in a starter kit is an adjustable wrench, so you'll want to buy one to add to this kit (you can find one for a couple of bucks at any of the big box stores).