If I give a company 8 weeks notice (my position requires lots of training) is that a bad thing?

Yes, a company can let you go once you have given them notice. The business standard is 2 weeks notice. I would suggest you give them 2 weeks notice and if they ask you to extend it longer you can make that decision when you are asked.

If you work in an 'at will' state and you have no contract with them, then they sure can fire you if you give notice!

Put yourself in your employer's shoes - you have an employee that could have access to all sorts of data and systems and who knows what else, and that same employee just told you s/he wants to leave you to work for somebody else, maybe even a competitor. Would you really want to keep this employee around?

Giving notice is just too outmoded. If you have your other job lined up, just go ahead and resign on the spot. Don't feel bad. This happens all the time and it's just business.

Yes they can & do, use good judgement.