Washing leaves some stains, which I want to get rid of.
It's easy to do with paper towel on the outside, but inside the glass it's a problem.
Wine Enthusiast (and probably many other places) sells a couple of sizes of very soft brush for cleaning wine glasses.
I'd say your best bet is a soft brush or maybe just a piece of soft foam glued to a stick. Nothing very abrasive, and squishy enough that you can reach any small nooks inside if there are any.
If your wine glasses are particularly stained trying soaking them in white vinegar for an hour or so before cleaning.
Clean the glass with warm water and a cloth (try bunching the cloth up in order to clean the hard to reach places, the vinegar should have loosened the stains). Another suggestion it to use a small scrubbing brush with a handle.
For a really good clean try boiling some water and putting it in a bowl. Hold the glass over the steam and use a cloth to clean/polish. I wouldn't recommend this for very thin glasses and you definitely shouldn't use extremely hot steam.
This is a great question. I have several techniques I use for this exact problem.
1) Try a wooden spoon with the paper towel wrapped around it. This works if you can reach most of it by hand and just need that little extra.
2) Bunch the paper towel in your hand so as to get extra length. Then use your fingers, extended, to move the bunch around the bottom of the glass. If this doesn't work, try number 3.
3) As your wife, girlfriend, son, daughter or other smaller person to help you, as they most likely have smaller hands ;--)