I, too, am a generalist, and have been for my whole 34 year career. I enjoy Blood Bank, but after our small hospital was bought by a large hospital group, we were trained to do the work the way the the group does Blood Banking. It was good to go through the training. It wasn't that everything was changed, it was that we were now going to do testing the same way as all blood bankers in the group. So, as slsmith said above, you might want to wait to see how your BB does things.
At our hospital, we run types, antibody screens, DAT's, fetal cell screens and crossmatches (electronic and serologic if indicated). Everything else is sent to the main central Blood Bank.
My advice to you is to figure out how you do testing (whether on your own or the "company way"), and always do it the same. Set up your tubes the same way every time; add reagents in the same order, etc. This way, if there is ever a question about your testing, you know that you always do it the same. Also, during training, don't just do the minimum. Test until you are comfortable with the process. You want to be confident in your work. Lastly, always take responsibility for your work. If a problem arises or your work is questioned, be ready to explain what you did. Knowing the procedures and processes will help you be able to troubleshoot these issues.
Good luck in your new job as a Generalist. As I said, I have been doing this for a long time, and I still love what I do!