Breeding budgies can be a taunting task especially for begginers. While they are probably the easiest parrots to mate it's still tough if you dont know what you are doing. I've written some easy to following instruction on how to successfully breed your pet birds ;-).
This guide can be broken down to the following:

-Selecting the Breeding Pair
While choosing budgies for breeding, it is necessary to give importance for their health rather than how they look. This is because only healthy budgies could produce the healthiest chicks. Budgies will be matured physically with 3rd or 4th months but it is not good to breed them at this age 

-Preparation of Budgies to Breed
Generally speaking, breeding is somewhat stressful for the female budgies as they have to lay the eggs. Moreover after the hatching of the eggs, raising the young ones involves both physical and mental stresses for both male and female birds as they have to make them ready to face the dangers ahead. 

-Provide Stress-Free Environment
It is very important to eliminate all the sources of stress from the environment of budgies. This is because, reducing the stress level strengthen the immune system of budgies and make them both mentally and physically good to breed. There are three main sources which cause stress: light, movement and noise.

-Good to knowThere are several things that you must provide your budgies with at the time of breeding. You have to make sure that budgies are in good condition both mentally and physically. Then you must provide plentiful of water and food, a nesting box and wait for some time to take the course. 

-Budgie Mating
The male budgie mates only if the female budgie is ready. Till then the male budgie spends a lot of time in courting her. While leaving your budgies to breed, the main thing which you have to do is to give time for them to court each other so that they become bonded.

-Laying and Hatching Of Eggs
 you are sure that mating occurred between budgies then be sure that the female budgie will be laying eggs between 10 to 14 days after mating. Budgies use to lay three to four eggs for per clutch and sometimes it may also lay around 10 eggs. If all the eggs are laid, then the female budgie sits on the laid eggs. After 18 days of incubation, the eggs begin to hatch and young chicks come out. 

-Development of Baby Budgies:
When baby budgies are born, they are naked, blind and helpless. After 10 days the feathers on their body start to grow and the chicks begin to open their eyes. In a few weeks, the chicks take a lot of foods, learn to move using their bodies and they squawk loudly. Around five weeks the chicks come out of the nest and try to feed themselves.

Source & Read More
Budgies are adorable as pets. They are very active, intelligent, excellent at talking, easy to breed and very friendly. Budgies natural environment are the Australian forests but none available in a pet shop are caught in the wild. Budgies through selective breeding have involved in two types, the English and the American budgie. Budgies are true parrots and develop many different colour mutations. Budgies can be tamed using the proper budgie training method and make a great companion pet. Keeping a tamed budgie as a pet is very fun and brings lots of enjoyment. Budgies are very responsive and intelligent and taming them is relatively easy. Below you can find some budgie training tips and procedure to help you.

The essential part of budgie training: Trust

Taming your pet budgie is simple but takes time and effort. The most important thing in budgie training is to make your budgie to trust you. The budgie needs to feel comfortable and relaxed when your moving around it. It’s ideal to talk and react slowly when you are around your budgie. Budgies are a small bird and you can easily scare them with your hands or even presence. Try not to make any sudden moves and always speak to them calmly when trying to handle them. You need to build a relationship with your budgie before making any training secession. If by mistake you make a sudden move or hit their cage let them calm down before trying anything else. As a side note keep in mind that owning and raising a budgie from young makes it the excellent candidate for budgie training.

The whole process can be broken down to the following parts:
- Trust Building (The Most Important Part)
Hand taming your pet budgie first in cage and out of cage
- Further Training outside the cage