Your budgies first moult will be around 12 - 14 weeks (3-4 months old). It can last for about two months (this is an estimation).
After that depending on your climate and where you live they can moult 2-4 times a year. Artifical air condition and heating does affect how your birds moult.
It is very normal to see a lot of white downy feathers and long feathers in and around the cage. As the new feathers grown in you will see what is called pin feathers where the tip of the feather has a small black sheath like covering on it. This will come off as the new feather will opens out.
During a moult you may notice your bird may become grouchy, tend to bite more and itch (it is not uncommon to see them rub their heads on their perches or cage bars). Again this is normal but if you find your bird is lethargic, fluffed up for long periods of time or not able to perch this would be cause for concern and you will need to schedule an appointment with your avian vet.