When the quarantine period of 4-6 weeks is over for your new budgie, you will want to keep both budgies in their cages and place the cages next to each other. This gives the two birds a chance to interact (at a safe distance) and realize that there is now a new budgie in the house!
After a couple of days with the cage next to each other (use your judgement for time), you can introduce the budgies in a neutral area. Neutral areas include a playground or playgym, someplace NOT inside of a cage.
Never introduce budgies by putting one inside the other's cage -- always introduce the two in neutral territory.
Putting a new bird in another's cage can cause tension and aggression between the two. The bird currently occupying the cage may become very territorial and try to protect his "home" from the "intruder".
Watch how the budgies interact.
If they are at all aggressive (blood being lost, feathers being pulled out), immediately separate them back into their original cages.
It is normal for a little bickering to occur.
There might be some squawking, chasing around, or odd budgie noises. To make things a little less tense, give them both some millet, and then reduce it to one stick to see how the budgies react.
If the budgies seem fine on the playground/neutral area, you can then move them both into the same cage.
If you move them into a cage previously occupied by one of the two budgies, make sure you rearrange it so it looks different and foreign to both birds. This will help prevent any territorial feelings coming from one budgie or the other.
The new cage should be as large as possible. At the very least, it should meet the minimum recommended size for two birds.
Include a variety of toys (ropes, bells, ladders) and perches (rope, swing)
Ensure you have separate food and water bowls for each budgie. Place one set of food and water dishes each on opposite sides of the cage
Once the two budgies are in their new cage, keep an eye on how they are doing. All new budgies will fight a little (but not too violently) and bicker to establish who is the "alpha" budgie. This 'bickering' should go on for at most couple of days, but then settle down.
If you notice that one budgie is agressively bullying or injuring the other, you will need to separate them back into two separate cages. You can try a reintroduction again a week or so later.
If you are interested in taming one/both of your budgies, you will want to work with them separately in quarantine and when they are together. After about a week of the two budgies being together, only then should you start training them. Work patiently with 'stepping-up' (even if one of your budgies has mastered that step when he was alone) and start back from the beginning of the "training book". It may be frustrating and your budgie may bite you (he's scared you'll separate him and his new friend), but keep with it! Once one budgie begins to show trust, the other one follow his example.