Will I be in a lot of pain after my hand surgery?

Depends. The amount of post-operative depends entirely on what procedure is being performed. Simple surgeries such has trigger finger release or carpal tunnel release are less painful than more involved surgeries. Also, pain perception varies from patient to patient.
Should not be... Hand surgery is typically not very painful postop. Elevation of the operated hand above the heart for 2-3 days will help. Many hand surgeons will inject a long-acting local anesthetic into the operative site at the end of the procedure and patients are sent home with pain medicine. If there is pain that is not relieved with these methods, your surgeon should be notified.
Not usually. Pain is not too bad after surgeries that do not involve bone. Carpal tunnel surgery typically makes the hand feel much better despite having surgery.