Build a really, REALLY simple computer system?

And does all that!

By build, you put it together, go get apps or write a few.
Bob

I can get good used android phones from Glyde that do this. While you may want to make this yourself, why burn your budget when for not much you can build a Rasperry PI model b+ and customize it as you wish?
Bob

When you say "build from scratch," I have to assume you mean that you want to purchase the components and assemble them yourself. Otherwise just get a cheap tablet, for which even the bottom-end machines can do the things you describe.

But if you actually want to build one, you will need, in addition to a set of precision screwdrivers and nut drivers that you can get from any computer store for about $5.00:

1. Case
2. Motherboard
3. Processor (CPU)
4. Power supply
5. RAM
6. Graphics card
7. Disk drive
8. Operating system

And, if you don't already have them:
9. Monitor
10. Keyboard
11. Mouse

That's really all you need. Online shops like NewEgg and TigerDirect sell what they call "Bare-Bones Kits" which provide everything you need.

Some caveats: The motherboard has to fit in the case -- an ATX motherboard will not fit in an mATX case. Your processor and RAM will need to match your motherboard sockets and slots. Your power supply will need to be adequate to power your choice of components (going too big just costs you money, though, and it's otherwise OK).

Assembly is easier and more straightforward than you probably presume. For the most part, the components all come with proper connecting cables included, and the little plugs are all either clearly labeled or only fit in one place. The only really tricky part is getting the CPU cooler (provided with the CPU) correctly bonded to the top of the CPU -- you need to get this right the first time. And one little surprise is the high amount of force it takes to close the little clamp that attaches the CPU into the socket.

If you build it, you will get a level of satisfaction you can't get off-the-shelf when you first power it up and everything works!

You might want to look at the "bare bones" offerings from Shuttle PC. I've put a couple of these together to be used with a school's projection cart. You can buy a low end for a couple hundred or so. For that, you get the box, motherboard and CPU cooling system. You add the CPU, RAM hard drive and optical disk drive if you wish. Of course you need to buy an operating system or use Linux. These are small and quiet boxes about 1/2 the size of a mid tower. There are smaller bare bones boxes as well but I've found the Shuttle manufactured offerings to be quite decent for what they do. A phillips screwdriver is about the only tool you need.

you would need to load an OS. The first generation of the PI's were most often used with Linux. The new one will run Windows 10 when it's released. The PI's were created to teach programming.