Compositing certainly works, and can be both photographically effective and cost-effective. That said, it's an awful lot easier to do if you can control the background when you're shooting; a plain, flat neutral background is a whole lot easier to extract a foreground figure from than something with detail or colours that come close to matching the subject. Luckily, you don't need much for this: white or grey seamless or even a flat white sheet will do the trick. You would have to keep an eye on the effect of light reflected from the background, though, and light the subject as you imagine him/her appearing in the final image. (By the way, this is pretty much how Joel Grimes creates all of his images.)
Unless you think you'd be a competent portrait painter, though, I'd steer well clear of the post-shoot lighting idea. Because that's essentially what's involved: using the photograph you made as little more than an underdrawing (and colour mixing guide) over which you would pretty much have to paint a new image. Even though dodging and burning (or using a grey overlay layer) gives you a bit of a boost over having to select or mix colours individually, you would still have to be able to create convincing modeling (highlights and shadows), and that's every bit the same skill in an image editor as it is with a paintbrush. It's even more difficult in one respect: you also need to undo the modelling that was present in the original image. If you have those kinds of skills (or are certain you can develop them), then go for it -- but understand that there will probably be an element of the painter's hand visible in the final image. For tweaks, though, there are plenty of tutorials on YouTube (I'd take a look at the FStoppersPPT channel to start, and keep an eye on the sidebar suggestions).
It's much easier to get the lighting right when you shoot. If you're thinking about a home studio or something similar, then you don't need to go to any great expense (at least at first). With any reasonably modern camera (one with a useful ISO setting at or above 400) a couple hundred watts worth of full-spectrum compact fluorescent lighting in homemade modifiers will do the trick until you can afford the "real thing". Two hundred bucks (US/CDN) and a bit of sweat equity will get you a rather sophisticated multi-light setup; for around fifty bucks and a half-dozen contributions to the swear jar, you can build a decent 30-40" (75-100cm) softbox and good-enough stand to get you going. And don't forget about the sun, windows and a cheap reflector or two (Peter Hurley built an entire career around a single window in his apartment before getting enough business to warrant a studio).
You can do your compositing entirely in Photoshop or the GIMP if you want, but if this is something you're going to be doing a lot of, then a good third-party masking plugin will make your life a whole lot easier. Two that I'm familiar with are OnOne Perfect Mask and Topaz Labs Remask 3. Both will handle things like translucency and hair/lace, things that will drive you bonkers doing manually. Both offer free fully-functioning trials (no need to spend anything making up your mind) and both have extensive tutorials on YouTube you can watch even before downloading the trials.