I would wait for at least 3 months before I take my baby on a flight. As far as I know babies dont have highly developed immune system, the only immunity they have is from the mother through the placenta which lasts for about 1 year. Breastfed babies get added immunity through breastmilk. So with the flu season in full swing I would not take a chance, dont want a fellow passenger coughing over my little one. Honestly depends on how much of a risk taker you are, its definitely easier than travelling with a toddler.
There is really no need for you to buy a separate seat for your baby till the time she/ he can fit in the bassinet the airlines provide. You could book a bassinet seat and put the baby in the bassinet in front of you, let the airline know that you have a baby travelling with you. You will need to buy an infant ticket the cost of which is lower than an adult ticket. However Infant ticket does not give you an extra seat.