So it begins, the love/hate relationship with hair. This is a part of self esteem where little girls of color learn to loathe or love their hair.
Learn to treat her hair like it is valuable/priceless. Take your time be gentle. Learn to love her hair. Check out Fotki and search Natural Hair, Nappy Hair, or Kinky Hair or go to U-Tube and look at this and other videos about hair care (this one was great): http://www.youtube.com/watch?v=cVgdf6W6el8
Separate her hair into four sections, then work at gently combing or brushing through hair from the ends working your way back to the scalp. Instead of vegetable oil, try extra virgin olive oil, jojoba oil, almond oil, coconut oil, shea butter with essential oils for a different scent (rinse out the olive oil but the others can be left in the hair. Other natural products which work lovely on the hair for conditioning are honey, mayonnaise, plain or vanilla yogurt. Also try Castille soap instead of regular shampoo be careful not to get it into her eyes though but if you must use a traditional shampoo, I would recommend Baby Magic Shampoo works well and has a pleasant scent.
Curly kinky hair can be dry so conditioning is very important. Hope this helps you and your new little one. Let me know and congratulations on being her mom.
Wanted to add a special note to you: "African" Braid shops are usually unlicensed and often braid hair too tight which can do permanent damage to the hair system and at night wrap the hair in a silk scarf or use a silk pillow case.