It truly is a blessed opportunity for not only the mother, but the infant as well, to be brought to the holy sanctuaries of Ahlul Bayt (a.s.). Nurturing a child the love of Ahlul Bayt (a.s.) by taking them to holy Shrines, and from such a young age, engraves faith and belief in their hearts.
Therefore, the mother should seize the opportunity of giving her baby the most exposure to the holy Shrines, and benefiting from that spiritual atmosphere.
That being said, like everywhere else one travels, certain precautions must be taken.
Try to select a date where the weather in Iraq is moderate. Not too hot, and not too cold. Spring, or Autumn, for example. Keep in mind that electricity is still a problem there.
Pack basic essentials for the baby's needs, from clothes to nappies. A compact pram would also be useful. Bring child medicine, like panadol, nappy rash, etc, and also medication for diarrhea.
Try to choose quiet times when you go for the Ziyarah, and avoid big crowds. Keep an eye on you child, and try not to be conspicuous. For your own protection, and for your baby's protection, pay sadaqah regularly, and keep a hirz on your baby.
It would be very useful to travel with people who would assist you throughout the trip. Try not to rely on others, but at times you will need help with certain things, and maybe just a little rest.
Continue to remind yourself that you wish to gain proximate benefit from Ziyarah, and that your intention is purely for Allah ta'ala. Also, remember that your reward is based on the amount of difficulty you go through, so everything you endure will be taken into consideration in the eye of the Almighty.
And please keep us in your duas.