Summarize:

The chick's parents will care for it as best as they can, but you should regularly check on the health and development of the chick. If the chick develops an illness, you may need to consult a veterinarian. By the time the chick is a few days old, it should show some bright colour, and should be beginning to put on weight. Generally a lighter chick is more disposed to illness or is being targeted by red mites.  If your chick doesn't appear to be developing or gaining weight, consider contacting your vet or handfeeding if you can. Once your chick has begun to develop feathers, and they have reached about half an inch long (approximately 1cm), they should feel strong when you touch them. If you think your chick is not developing properly, you should consult a vet for some expert advice, but there are some basic checks that you can do at home to help the chick develop healthily. The first of these is to look under the top mandible (beak). If you notice a build-up of food, this can lead to an undershot beak.  If you see some food stuck there, very carefully remove it with a toothpick. If you think that your chick is beginning to develop splayed legs, try adding some extra bedding to the nest.  If the chick cannot stand up straight, perch, or walk around easily, he may have splayed legs. Look to see if the legs lean out to the sides rather than underneath the body.  If you are uncertain or you think your chick is ill, don't hesitate to contact your vet. Once the chick starts coming out of the nest on his own, you can help the process of weaning along, and then help him leave the nest. Once you notice the chick coming out of the nest, you should provide a shallow dish of food and place it at the bottom of the cage. The chick will start eating the food from the dish, beginning the weaning process.  You should allow the chick to wean by itself, but monitor it closely to ensure that it is eating enough.  Provide a dish of fresh water so the chick can learn to drink from it. Monitor the chicks and parents; when they have not been fed by the parents for at least a week, have not asked for feeds and you can see they are cracking seeds and filling their crops well you  can considered them weaned. Once a chick is weaned you can remove him the from the nest. This will help his development and the development of any younger chicks in the nest. Put the weaned chick in a large cage that you have set aside for young birds. Ensure the cage has plenty of food placed in different areas, but especially in a dish at the bottom of the cage, and a supply of fresh water.  Keep a close watch on the young birds and check that they are eating sufficiently. Make sure the bird's crop is full before covering the cage for the night.  The crop is the place where budgies store food. When it's full, there is a noticeable lump on the front of the chest. Once your chick is developed and independent, you can book him in for his first check-up with the vet. Bringing him to the vet will reduce the chances of a hidden problem developing. The vet will also be able to give you some good tips on how to care for a young bird. Before you go to the vet, it's a good idea to prepare yourself well.  Write down a list of all the foods and supplements you are feeding your bird. Be able to describe the bird's living environment. Bring in photos if possible.
Monitor the chick in the first days. Check the babies for fixable developmental problems. Help the chick wean. Remove the chick from the nest. Consult your vet.