Electrical problems you say... hmmm, sometimes they can be hard to diagnose. However, you should be able to get to the bottom of it with a little tenacity, patience and beer (scotch, wine, or any other spirit will do). And even though you say you are not well versed with a multi-meter there is still hope... they are not too difficult to demystify.
Okay... with your demonstrated will to get to the bottom of this, might I suggest you read
this thread of mine from recent times when my own '03 had charging problems. But first... first MAKE SURE that new battery of yours is FULLY charged before you proceed. You see, an undercharged new battery may not ever fully charge unless it starts life out completely topped up.
Back to that thread o' mine... it is chocked full of insight and suggestions from many here who know a thing or two. I personally learned a bunch on the topic of charging systems and although I too am a novice at this kind of thing, I managed to unearth the problem under the guidance of many here. During the whole process I also managed to improve on the charging system making it better than it was from the factory.
The voltage readings you provided are curious. If under higher RPMs the voltage actually DROPS, in raises eyebrows. It could be the regulator/rectifier, or even poor connections somewhere. Often one of the bad connections is the ground wire attached to the frame. So... start with battery connections and go from there. Follow the leads from the battery terminals and look for corrosion along the way.
The first thing I would do is charge the battery again to a good and healthy FULL charge. Then run a couple of voltage tests to see what you get. Good luck and let us know how you make out!!
