If it is critical for you to have a house-trained animal, birds may not be for you. Most birds who do well in captivity eat seeds. However, if you are intent on doing this, you might want to select a larger bird like a parrot or a cockatoo.
Larger birds have slightly lower metabolisms and pass food a little more slowly.
They are also pretty smart. You'll need to have perch sites in each room and place newspapers below each. Reward the bird with food, love, and praise when it eliminates on the paper. Since you ask about house-training, you may not be thinking of caging the bird. There should be a refuge, a large cage or a perch, where it can stay safe.