How to obtain a Clean yet Distorted guitar sound?
At the risk of being a bit of a gear snob: my first amp was a little peavey solid state like the bandit. When I finally saved up for a fender all tube combo the difference was amazing. Not to say you have to go get a new amp, just that you can go crazy with chasing tones while your fundamental gear is holding you back.
What are some of the things that influence a player's "tone"?
It's in their fingers, by which I mean the way they attack the string, the way they use vibrato, the inflections they use. All of these things contribute to tone in subtle ways.
What influences a player's tone has a lot to do with the equipment they use. A Strat sounds slightly different to a Tele, and a Les Paul sounds drastically different to a Rickenbacker.
Guitar - How to switch from clean to hi-gain
You can get (mostly) clean tones from a high-gain amp setting if you go into the amp with a very low-amplitude signal. To put it simply, the amp has a maximum signal level that it can handle cleanly; beyond that, it goes into saturation and distorts the sound - the amplitude (volume) doesn't increase anymore, but the tone changes to "crunchy" and then "hi-gain". The "gain" knob determines by how much the amp multiplies the input signal from the guitar; so you can compensate for a higher "gain" setting on your amp by turning down your guitar (putting aside stuff like a higher noise level for now).
What factors contribute most to the tone of an electric guitar?
In clean guitar, the pickups and guitar build are a big deal. A hollow-body jazz guitar played in the neck position is a different animal from a Telecaster played bridge. The different tones are loved for what they are, and not just as inputs to a distortion circuit. However, clean tones still greatly depend on the amplifier. The output of a passive pickup does not sound good if it is amplified using a hi-fi amplifier and speakers. And of course, clean tones are influenced by effects. Reverb does a lot for clean tones, recreating the ambience that is missing in the electric guitar (particularly solid-body). A popular effect applied to clean guitar is compression. Compression is the key element in some clean tones, which depend on it as much as heavy styles depend on the distortion.
How important is brand name and type of string to the overall sound of a guitar?
My two cents is that I believe I can feel when I have D'Addario because they feel better than Ernie Ball Slinkys. I don't believe for a moment that there's a profound difference in sound, though.
How do I get this tone
Bear in mind also that the way the string is plucked, and where it's plucked will make a marked difference to the sound, no matter what effect is used.
Work on the clean tone first.