Both GR and Newtonian physics use coordinate systems and in much the same way, except that the coordinate systems in GR are generalized coordinates, and the coordinates in Newtonian physics usually are not. One COULD use generalized coordinates with Newtonian physics, but the math required to deal with generalized coordinates is not taught in high school. It's usually taught at the graduate level in college. A bit of the math needed to deal with a few special cases (such as polar coordinates) is snuck in a bit earlier, not very rigorously, but it's not very rigorous nor does it handle totally arbitrary coordinate systems.
One of the things this means in practice is that a change of a coordinate by one unit in GR does not in general (or even usually!) represent a change in distance of one unit - coordinates do not directly measure distance. This is the easy part, sometimes it seems that attempts to explain this fall on deaf ears, and I'm not quite sure why. The procedure for getting distances out of coordinate changes in GR involves using the metric.
Some authors banish the idea of the observer from GR, calling it "confusing" and ambiguous". I'll give a quote from one such paper by Misner, "Precis of General Relativity",
http://arxiv.org/pdf/gr-qc/9508043v1.pdf
A method for making sure that the relativity effects are specified correctly
(according to Einstein’s General Relativity) can be described rather briefly.
It agrees with Ashby’s approach but omits all discussion of how, historically
or logically, this viewpoint was developed. It also omits all the detailed
calculations. It is merely a statement of principles.
One first banishes the idea of an “observer”. This idea aided Einstein
in building special relativity but it is confusing and ambiguous in general
relativity. Instead one divides the theoretical landscape into two categories.
One category is the mathematical/conceptual model of whatever is happen-
ing that merits our attention. The other category is measuring instruments
and the data tables they provide.
The reason for this banishing of the observer suggested by Misner is explained further in the paper where he explains what he calls "the conceptual model". The reason he gives is basically that space-time is curved, and that the curvature gives rise to effects are not intuitive. In fact, in the presence of curvature, the idea of coordinates remains the same (contrary to the original poster's question), but the idea of "observers" in "frames" becomes a bit harder to define. I would not say the concepts of "an observer" or "a frame" are impossible to define, personally, but I find that by the time the definition is precise enough to be useful, it's lost 95% of the audience :(.
What is the conceptual model? It is built from Einstein’s General Rel-
ativity which asserts that spacetime is curved. This means that there is no
precise intuitive significance for time and position. [Think of a Caesarian
general hoping to locate an outpost. Would he understand that 600 miles
North of Rome and 600 miles West could be a different spot depending on
whether one measured North before West or visa versa?] But one can draw
a spacetime map and give unambiguous interpretations. [On a Mercator
projection of the Earth, one minute of latitude is one nautical mile every-
where, but the distance between minute tics varies over the map and must
be taken into account when reading off both NS and EW distances.] There
is no single best way to draw the spacetime map, but unambiguous choices
can be made and communicated, as with the Mercator choice for describing
the Earth.
