How long should it take a cut on my finger to heal? I cut it in the kitchen last week and it is still a bit tender and sore.

Weeks. It truly depends on the severity of the cut (superficial vs deep) and how much protection that site is recieving --> is it constantly being irritated or are you protecting it with a bandage. Furthermore, if it is draining pus and looks reddened/swollen, you may have an inflammatory reaction going on due to presence of bacteria. Keep the area clean, protect it and it should heal.
Maximize healing. The real question is what not to do that slows down healing. Avoid using alcohol, peroxide, hibiclens, betadine, and bleach solutions. Inflammation ( red, hot, swollen, tender) is important to healing so anti-inflammatories (motrin, advil) are bad. Cover the area with a clean dressing changed every other day and protected from trauma. Antibiotics do not help healing unless it is truly infected.
2 weeks for most. Keep it clean and prevent infection. The time it takes to heal depends on the location (crease) and depth. Usully 2 weeks.