Shards do not have a mind of their own, only as Sazed says "a weak will". (Or something like that) They want to follow their intent and will actively seek out a person to hold their Shard and follow that intent. If you remember back to the end of TWoK Dalinar realizes that Honor was not answering his question about Sadeas, he just assumed that he was. Honor cannot hear him, and Tanavast is dead. Read Dalinar's last chapter and the vision where he fights the Midnight Essence (I'm pretty sure that's where he aska about Sadeas). Once you read it, knowing that Honor isn't answering his question, it all makes sense.
As for why they have no mind but Syl does, that goes into some Realmatics, at least in my theory. I'll explain it, but since you're relativesly new, like you said, I don't know if you know about Realmatics yet. If you do, great! If not there's a super awesome Shardcast here to help you. Remember that everything I say below could be completely and utterly wrong.
This whole theory hinges on how I view the Congnitive Realm. I think that it is a sort of overlap between the Physical and Spiritual Realms, the place where they interact. An object needs to have both Physical and Spiritual aspects for a Cognitive aspect to form, in my opinion. So a Shard, comprised mostly of spiritual energy, has very very little physical to it, at least in proportion to the massive amount of spiritual energy they contain. So not much of a Cognitive aspect ends up forming, nothing beyond that will to follow its intent. But, once it picks up someone to hold it, that person brings enough physical, as well as their own Cognitive, to sort of complete the entity that is the Shard, and allow its full power to be used.
Now Syl is much the same, almost a like mini Shard. (Or a Splinter to use the right term) She has some spiritual but almost no physical, allowing her to barely manifest and affect the physical world. But then the Nahel bond forms between her and Kaladin. She starts getting a boost from his physical aspect, allowing her mind to begin to work again. In return, he receives a little bit of her spiritual power/connection to Honor (I'm not yet sure which) which allows him to begin Surgebinding. This is all part of a theory I've yet to post in its entirety, but does this bit make sense?