I may make my own emulator some day for educational purposes, and it will probably be in C++. My question is, What graphics engine do you guys use and how would you get it? I don't want to have to learn x86 assembly, so I would simply use someone else's. Thanks. ^_^

perhaps the easiest way (if your using windows) is to use the WinAPI call to draw a pixel (on a fast machine this should be ok for your own use) or absolute speed you should use directx - though this is harder to learn and you would need to find tutorial on directdraw/3d (these are integrated in version 8+).
I suggest drawing pixels instead of some graphics engine yourself because you should use a scanline based screen draw instead of a bitmap based one (used in most 2d graphics engines)
p.s. you should't need to touch assembler as c++ is sufficiently fast for todays PCs (if you're running slow its more likely assembler would't make much difference anyways)

Just my two cents about graphics. WinAPI is slow and deprecated. Additionally it is MORE difficult to learn the WinAPI than it is to learn DirectX. I would recommend DirectX if you are looking for a good balance of ease of use and performance on Windows. If you plan on porting to any other platforms (or you care about your soul) I would recommend using SDL, the Simple Direct Media Layer which is highly portable, and uses DirectX as a backend on Windows.

I was referring mainly to doing it in DOS. I prefer DOS (or DOS-style) emulators over Windows ones. I hate doing it in Windows, because it feels like it's invading your game and disturbing you. It tends to make the game less enjoyable for me. So...

In directx you can program with barely touching windows (as most games do)(only requires basic initialisation) in fullscreen - so its like dos, but if you really want to use dos I would say to use the allegro library as lots of PD dos games use this as well as some emulators, and I believe it has a windows version for when the inevitable happens and microsoft drops the lousy dos support that xp has, I think there's not much other than allegro (free that is) besides writting the emu in real mode and programming it that way (there's a qbasic and turbo pascal emu's written this way)