I would recommend a desktop computer of some kind, unless a laptop is required.
I am currently attending art school for visual effects (3d computer graphics for film) and I use a custom built pc desktop. For 3D rendering and video editing, the most important specs for a computer are the RAM (amount and speed, 16GB is preferred), the processor (cores and clock speed, quad-core or better is recommended), and the hard drive (space and speed, a quick 3.5" drive at 1TB or more is recommended).
The graphics card isn't quite as important unless you're gaming. Any mid-range nVidia quadro or geforce card would work (quadro 2000 or gtx 560 or better) just fine. But you may not even need a card that fast.
As for a monitor, I definitely recommend looking into more color-accurate monitors. I use a Dell U2410 and it is great! The U2412m is also good I hear.
My entire setup is probably around $2000 but you could easily sacrifice in some areas to get it down to $1500 or $1200 if needed.
If you want to buy a pre-built computer, make sure you get at least a quad-core processor, 16GB of ram, and 1TB of hard drive space with any kind of dedicated graphics card, and it should work for a while. Same with a prebuilt laptop, but it will probably be more expensive than the desktop equivalent.