(If nothing else, this will give you good insight into what it is like to be a man.) While all of this sounds like it might be too difficult to deal with - and many women will come to that conclusion - it is worth pointing out that if you can navigate through the difficult aspects of attracting and dating a shy man, he is likely to be completely faithful to you.Due to the lack of female attention he has attracted or received, he is like an emotional spring that you will release.
As I said, he will probably start to do this on his own anyway, so rather than forcing it, just be cognizant of the transition.
That offense will reflect negatively on you and the relationship.
So be careful about how much (and which) initiative you take.
If they have even the slightest doubt about your intentions they will hold back.
So whatever you would normally do to indicate your interest to a confident man, double or triple it for a shy man (in proportion to his shyness): While I don't suggest that you overtly ask out a shy guy on an "official" date (this would too flagrantly undermine what he knows to be his role as described above), I do suggest that you suggest and then ask him to hang out under casual pretenses.
Once this happens, you will need to gradually assume a less assertive (more traditionally female) role, because he will then be more liable to get bored with you, wonder if he could do better, consider his other options (the perception of which his newly-found confidence will likely inflate), fool around, etc.