There are two routes you could take…
1- An ENG style camera with built-in zoom lens and XLR audio inputs.
Given your budget I would look at something like this:
Canon XA20 Professional HD Camcorder : $2,200
Or a Panasonic HMC-150 camera, which can be bought used for about $1,800 or new for about $2,650.
2- A DSLR camera which will have interchangeable lenses, shallow Depth of Field ( your subject is in focus but the background is blurred ), and generally produces an video image that looks a lot like a motion-picture film camera.
Given your budget, the best DSLR camera for video is the Panasonic GH3 camera, which records full 1080 24p/30p/60p video at bit-rates up to 71 Mbps. The GH3 can also shoot with almost any 35mm camera lens made, including lenses made back in the 60’s and 70’s which are dirt cheap on eBay. ( you will need the appropriate lens adapter, which are fairly easy to find )
The GH3 also has the lowest rolling-shutter timing of any DSLR, so that fast pans won’t look as warped as they do with other DSLRs.
The GH3 also has unlimited recording time, provided your batteries last and memory cards can hold the data. It’s possible to shoot one 12 hour take with a GH3 using a large SDXC memory card and an external power supply. Most of the other DSLRs can only record 30 minutes or less for a single take.
If photography is more important to you than video, then you might look at the Nikon D7100 which records very good video and is an excellent still photo camera. ( the GH3 takes very good still photos, but it really excels at shooting video )
So which camera to buy?
A lot depends on your shooting style and what you like to shoot. For live event work an ENG camera is usually the way to go, but for documentary and film-like footage a DSLR will produce the highest quality image.
I shoot with both, but most of the time I shoot with a Panasonic GH3 camera. ( I use an ENG camera for live events )
Good luck!
– Guy