Say Hello to who? It really makes a difference. Professionally, if I'm being introduced for the first time, I'll extend my hand to shake. If someone else is being introduced to me, I'll wait and expect them to do it but will offer mine if they seem hesitant. If it's a collegue I've met before, we usually forgo handshakes and just go with "Hello/Hi; nice to see you again".
Personally, again it really depends who it is. Is it someone I already know? It could be anything from a "Hey" to a full-on bear hug. Being introduced to a new person very informally, I would not expect to be shaking hands and would not automatically offer, but would accept a hand if it were extended.
As to your other point - there is little more off-putting than a limp handshake.
For Pete's sake, parents - teach your girls and boys how to shake a hand. If you're an adult and lack this skill, find someone to tutor you - quick!!