I decided to stick with durable items rather than consumable products, so you won't find tape, glue, sandpaper, pencils or fasteners, even though all of these really are tools.
For the record, I would put duct tape, masking tape and electrical tape in my first tool box, along with super glue and urethane glue, a few sanding sponges of varying grit and a medium sized fastener assortment kit that included screws, nails, wall anchors, etc.
What about saws, squares, chisels, socket wrenches , or any of the thousands of other tools that you would have included? Most of those are designed for more specific tasks like woodworking, auto or electrical. I picked versatile starter tools that offered good bang for the buck.
There's a reason this tool's first name is 'utility.' It's just plain useful. Ask most survival experts what they would want if they were stranded on an island with only one tool, and I bet every one of them would say, "a knife." A utility knife is just as important to a handyman's survival. You'll want to pick one that uses standard utility knife replacement blades and makes it easy to change them. A good grip is essential.
Let's see... cut, scrape, clean, pry, chisel, spread, pound, lift - to name a few. I'd say "5-in-one" is a conservative estimate of the uses for this tool. There might be a more versatile tool to put in your toolbox, but I haven't found it yet. Get a good one with a sturdy blade and a padded handle. It's worth the extra couple of bucks.
Most homeowners like things to be plumb and level - that's horizontal and vertical to most of us. You could just eyeball it, but a torpedo level makes it easy. Most available now have at least 3 vials for different orientations. I like an open vial for easier viewing and a magnetic strip for additional convenience. Some people prefer aluminum frames, but I think plastic is fine.
Another pair of pliers? Absolutely! Long nose pliers, sometimes referred to as "needle nose," are great for working in tight spaces. They're like an extra hand with tiny, strong fingers. They can be used to wrap wires, retrieve screws that you dropped, hold nails while you set them, and on and on. Choose a pair that is 6" to 8" in length, with a wire cutter and insulated padded handles.
Sometimes called a strap clamp, this is one of the most versatile clamping and holding devices you can own. It can hold a wider variety of sizes and shapes of objects than anything outside of duct tape, and you don't have to clean off any sticky residue! Use it while you're assembling, repairing, screwing, and gluing. Choose one that is 1" wide and 10' to 15' long. Removable corner grips make it easy to assemble and hold mitered corners.