A laptop is not advised; they have a small thermal envelope, and are significantly more expensive. They also limit your options for capture devices.
Desktop, advise at minimum an Intel Core i5 and a GTX 400+ or RadeonHD 5000+. Real-time video encoding is an EXTREMELY 'heavy' process, computationally speaking. Doesn't matter if you're streaming consoles, encoding video takes a lot of work. And OBS requires DX10 hardware support as it uses the GPU for scaling, compositing, and a few other things on the back-end.
For a capture device, would strongly advise a Micomsoft SC-512N1-L/DVI. It's what I use, and is the best capture card on the market for livestreaming purposes at present. Added bonus, it'll accept signals directly from all of those (aside from PS3, which uses HDCP unless you break it with a few specific HDMI splitters, or send it over component cables). Does HDMI/DVI with embedded audio, component, VGA, S-video, composite, and even RGB if you have an rgbmodded retro console. It's kind of pricey at $330, but it's worth every penny compared to the Avermedia crap. Can also get one from SabrePC as the Yuan SC512 for $200, but they're having trouble keeping them in stock so it can be a wait.