I used a Gaggia Classic for about 6 months before upgrading to a Brewtus III. I did have a little trouble when I first got the machine but it's all about finding the technique that works best for you.
When steaming I would remove the plastic sleeve but I would keep the smaller plastic screw on part attached to the wand. When I start steaming I made sure to keep the wand shallow, very shallow. With a stock Classic steaming wand I would recommend using a small milk pitcher (I would 1/2 fill a 12oz pitcher). Make sure you purge the wand before steaming milk.
As for your issue with low movement in the pitcher...
There are two things that I can think of, one is too much milk. This is de-bunked when you said the milk heats quickly. The other thing it could be is just technique, particularly the angle of the pitcher. Pour 6-8oz of milk in a pitcher and steam it, trying different angles to get the best movement of the milk. Take 15 mins and a pint or two of milk to experiment with. Practice makes perfect and it just sounds like a bit of practice that is in order.
Acceptable milk is possible with a Classic
http://www.flickr.com/photos/robsonbarista/3555599342/Good luck!