Is it permissable--or advised--to follow IFR routes while flying VFR to get an idea of in-route IFR?
Yes.
Would doing so while under flight-following by approach be considered an annoyance to them?
Absolutely not. Radar flight following in terminal areas is generally encouraged for collision avoidance reasons.
What about, using the same sort of scenario, flying various IFR approaches while VFR to get a feel for those?
No problem from an ATC standpoint (see below!) -- just follow the procedures for "Practice Instrument Approaches" in
AIM Section 4-3-21.
At non-busy controlled fields, might asking to fly an approach add a burden for the tower?
If they're too busy to handle practice approaches, they'll tell you. If they don't, it's perfectly OK.
That said, as a working instrument instructor for
Professional Instrument Courses (the 10-day instrument folks), I have a couple of other points that may help you get the most out of your IR training.
First, while getting flight following on every XC flight is a good idea (it gets you a lot more comfortable on the radios), I would advise you not to try flying instrument approaches on your own before you are trained on them. You stand a very good chance of doing them improperly (and that
will annoy the controllers) as well as making habits of improper or inappropriate procedures or techniques (which will make your instrument training longer and more difficult as your instructor will have to break you of them before teaching you the proper procedures/techniques).
Second, the most important thing you can do to prepare for instrument training (other than really learning the ground training material, not just studying the test questions and answers) is to develop your mastery of the aircraft. Practicing precision landings (touching down
on a preselected runway centerline stripe with the stripe between the mains with no bouncing or drift at touchdown) and steep turns (stable turns with no roller-coastering or wandering bank angle, and I mean
stable -- just staying within 100 feet of altitude or 10 degrees of bank isn't enough) are two ways to significantly improve your control of the aircraft. I realize these may not seem to be related to instrument flying, but trust me, the skills they develp really are important.
And if you really want a control-developing exercise, try putting the aircraft in a standard rate turn and holding it there at about 10 knots below max flap speed while you go from clean to gear down (if you have retractable gear)/approach flaps, hold it in the turn in that approach configuration for 15 seconds or so, then back to clean for 15 seconds or so, and continue the cycle. If you can do that consistently with stable altitude and steady airspeed, you will be a long way towards real control of your aircraft.
Good luck and have fun!