Having a DNA test done on your bird
This is the most reliable method, and actually some types of birds and parrot, this is the only real method to tell what is the gender of your bird.
However, when it comes to budgies, not all the people are willing to have a DNA test done on their budgies, and there are other fairly reliable methods you can do to determine the sex of your bird.
If your budgie starts laying eggs
Well, this one is pretty straight forward, we all know females are the ones laying eggs.
So if you bird starts laying eggs, you can rest assured it is a female.
Sex linked mutations
Ino Opaline Cinnamon TCB Slate
All sex linked mutations can give you clue, we are talking about x-linked mutations where the female either has the mutation or not, but cannot be a carrier.
If any of the offspring has sex linked mutation,then you can simply look at the mother (hen) and see, if she doesn't have that mutation then the chick is 100% female.
Looking at the budgie's cere
This is usually the way to go in order to know the sex, if you have a male or female budgie
The reason being: in most cases it is easy to determine the gender by just looking at the cere.
use the ceres image for reference
In general, adult males have deep to royal blue cere color.
And, adult females have tan blue / white cere color or brown (usually crusty). Depending if they are in breeding condition.
Of course it still depends on the variety and your bird's age.
In younger budgies, before they reach adulthood, which is about the age of 5 - 7 months, have different cere color.
In general, young males have a more even pink/ purple color across the cere, while young females have white or pale blue color with some pink/ purple around the edges.
Read this article for more about identifying your budgie's age.
Behavior of your budgie
This is the least reliable method when it comes to determining your budgie's gender, and here's why.
Budgies have so much different and various personalities, so a lot of the times it could be misleading.
But with that being said, sometimes it could be helpful to give you a hint, or a direction to what might your budgie's gender be.
Males are known for being much noisier than female budgies. They usually chirp or sing and for long a very long time periods.
While female budgies make sounds too, but it is usually more of an aggressive sounds, and not for a long period of time.
Males also bob their heads up and down or tap their beaks against their cage or other budgies.
Males also regurgitate food, this is actually one of the signs to show females they are healthy and capable of bringing food to her so she could take care of the babies.
However any of these behaviors, could be done by either males or females, so it's not really reliable, but if you had to guess, there is a high chance you heading to the right direction.