It sounds like your daughter is showing feeding readiness signs, Michelle. If oatmeal is giving her a problem, change to another form of cereal, such as rice or barley. The only real reasons to give infant cereal are to teach a baby how to transfer a solid food to the back of the tongue and to introduce iron into her diet, as it's thought that a baby's iron stores begin to deplete by 6 months. Actually, at 6 months, any infant cereal should be rather thin, anyway, so there's no rush to thicken it up.
Since your daughter seems to be tolerating it well, other that the constipation, you can start introducing the veggies and fruits. Try a new food for three days before starting another so that if she has a sensitivity, you will know which food is triggering it. Finally, if your daughter is on formula, there's already iron in the milk, so she'll probably be fine. If she's breastfeeding, you might want to discuss with her physician using a small iron supplement if you choose not to feed cereal.
Our parenting advice is given as suggestions only. We recommend you also consult your healthcare provider, and urge you to contact them immediately if your question is urgent or about a medical condition.