Yes, budgies can live with cockatiels!
In fact, you can put as many cockatiels and budgies you want in a big cage. But make sure you keep an eye on those cockatiels and budgies, there could be a fight.
And yes finches can live with a cockatiel too, lovebirds can too. They get along quite good but like I said just in case I would watch them to make sure they get along well.
If your thinking about getting some new birds at a pet store or breeder, you should ask him/her if they are girls or boys.
Remember, you might dont want babies!
Hope you have as much luck as you can get,
And have the happiest day of your life.
Owner of a cockatiel and loves birds such as budgies, lovebirds, finches and canarys!
There is an old expression: "Birds of a Feather Flock Together".
In the context of this question, it means that birds are happiest when they are in the same area or cage with birds of their own kind.
However, unlike fish, the personalities of the birds may come into play. If the birds were born and raised with a variety of birds, then there shouldn't be a problem. But, if you mix the birds from retailers or friends who did not get them from the same source, then you might be asking for trouble.
Also, Budgies have been known to peck at Cockatiels' feet, for some reason (not exactly where the phrase "pecking order" comes from, but, you get the idea!).
Rather than rely on the responses of strangers, why don't you simply ask your local veterinarian (who, presumably, you're going to take the birds for health checks) or pet retailer (not the department store ones; they don't know anything about pets!) or go to the library and do some research on the subject?