[bmurnahan]

I thought playing the scale of the key would be the way to go about improvising or lead guitar

This can only take you so far. Even if all of the chords come from the same key certain notes just do not work well against certain chords. The note C against a G chords for example just doesn't work well at all.

As long as you are moving in a stepwise fashion through the scale at a decent tempo the notes do not have a chance to create these clashes. It's only when you stop on a note that you have to be careful.

To try some melody over the chords should I just play with the Pentatonic Scale in mind and move the root note to C and play that over the whole progression (which is what i've been doing). Do I move the scale to the root note of the chord being played?

Each of these approaches can work. With the first one you have to be careful (see above).

With the second approach, assuming you just use pentatonic scales, it's not much different than just playing through the chords since the pentatonic scales are made up of the 1, 3 and 5 of the chord plus two great color tones in the 6th and the 9th.

Or do I find the individual notes in each chord and doodle around with those?

Playing this way you will always be safe.

Is there anything particularly limiting with playing scales?

Only having one way to approach anything is limiting.

I'd really like to start playing Jazz and I keep hearing how 'following chords' are the way to go with improvisation in that genre.

All great jazz players follow the chords. A good working knowledge of chord tones is the most important thing that you can have. Different scales are valuable because they provide different color tones to fill in around the chord tones.

They really work hand in hand but you really have to be aware of each chord that you are playing on as it goes by. Learning to improvise takes time and there is always more to learn. Be patient and have fun.

Bob