Look into an HDSLR or a mirrorless camera that shoots well at both.
Here's some suggestions for setups.
Nikon: D5100 if you're cheep, D5200 if you have a little more. The D5200 has awesome video quality. D7100 would be a great option as well if you want to spend a bit more, and be able to use the older Nikon Autofocus lenses that have aperture rings.
Canon: T3i if you're cheap, 70D if you want a slightly better DSLR, and a 7Dmk2 if you're going to shoot a lot of sports.
Lenses? a Tokina 11-16 or 12-24, and either a 17-50 f2.8 from Tokina/Tamron or a 24-70 f2.8. Vintage lenses are awesome for both. Nikon can only really use Nikon and Leica R, Canon can use Nikon, Pentax, M42, Olympus, Yashica, Contax, etc.
Panasonic: GH2 if you're cheap, GH3 for the best value, GH4 if you want 4k video, and such. I'd recommend the GH3.
Caveat? the GH's are not the best for photography. They're definitely optimized for video. Pick up a cheap used Olympus E-Pen camera along with a GH# for 100-150 for something better for travel/street photography.
Olympus: OMD. Awesome Do-all Mirrorless camera. Not quite as awesome for video as the GH cameras, but far better for work with stills.
Lenses - 14mm f2.5, 20 f1.7, Vintage lenses - any old SLR/Rangefinder lens will be adaptable.
Sony. A7 series. They're all pretty amazing... They're a bit more pricy though, and not the greatest for stills in my opinion, but still better than the Panasonic GH IMO. They're full frame, so they will be harder to focus on your subject at wider apertures, but you can also get ridiculous depth of field, even with wide lenses.
Caveat: Sony E-mount full frame lenses are expensive. They're awesome, but they cost an assload because there's not much of a used market for them, and . I'd go with vintage lenses and focus on video if I went with the A7 route.
Recommendations: Vintage zooms: Nikon 35-70 f3.5 ais, 70-210 f4 series E or the 75-150 f3.5 series E. (you don't need one of the big 70-200 f2.8's they're huge, heavy, expensive, and not any much better in IQ when shooting video)
Otherwise, go with vintage primes. Konica and Minolta lenses are ridiculously cheap as well and are worth looking into adapting.
With any/all the rest of the money you save..
- Audio gear. A microphone is key.
- A stable video tripod (the Ravelli is a great option for not too expensive of a price) Add a photography ball-head so you may switch between the two heads whether you're shooting video or photo.
a glidecam would be pretty awesome for mobile stabilized shots, the HD2000 would work for pretty much all the aforementioned cameras - though the smaller DSLR's and the GH cameras might be a bit too light without larger lenses, so the HD1000 would work well.
Then get a quick release plate for the glidecam that would match best with what tripod you get.
From there, you should be good. I'll let the rest of the peanut gallery chime in from here.
Have a nice day, Bluename.
I am.. the LAW!