Linnies are awesome little birds, and can learn to talk as well. I have a female lutino named Peep, and she is nothing but sweetness and play! The males are more talkative, but my little girl could say "step up" when we brought her home from the bird shop. I really can't say enough about how much we love our little linnie! Here is a pic of Peep.
As far as parrotlets and budgies, I can't say from personal experience, but I think most all parrots are trainable. As they say, there are no bad birds, just bad bird owners. What I'd do is visit some at a bird shop or breeder in your area to get an idea of what each one is like.