You asked

Is it normal for my baby to vomit?

As your infant adjusts to life outside the womb it is normal for them to vomit. However, before you move further you need to determine if it is vomiting or if they are just possetting, bringing small quantities of milk back up.
When your baby is vomiting, a lot more will be coming out and they are likely to cry as it can be an upsetting time for them.
There are many reasons that you little one may get sick and they don’t all necessarily mean that your infant has a stomach virus. Car sickness, indigestion and coughing too much can cause your newborn to bring up their dinner. Food allergies can also be a cause of vomiting so if a particular food keeps making your baby ill, stop giving it to them and take them to their GP.
However, If your little one has gastroenteritis, an ear infection or a urine infection their vomiting will be sudden and is likely to be accompanied by diarrhoea.
Unfortunately, a vomiting episode can last anywhere between six and 24 hours and while there is not much you can do during the time, you need to ensure that your youngster stays well hydrated.
When your baby is dehydrated they will have a dry mouth, sunken fontanelle, fewer wet nappies and may be floppy. If your infant is showing signs of any of these it is important that you up their fluid intake.
Fortunately, there is usually no need to bring them to the doctor; however, you should make an appointment if they are vomiting for more than 12 hours, have a skin rash, a bulging fontanelle, are experiencing shortness of breath or if there is blood or bile, a greenish liquid, in their vomit.
Always trust your instincts and if you feel there is a need to take them to the doctor do so, it will help to put your mind at ease.
Once you baby has not vomited for more than 12 hours you can start to ease them back onto food, making sure they get extra milk or water until they are fully better.

More questions

Firstly, you need to determine if it is vomiting or if they are just possetting, bringing up small quantities of milk. 
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