I have to say I agree with illusive. Sounds like you have enough on your plate already at this time.
For a bird to be "quite affectionate" it takes a LOT of time, patience, and consistency. To be quite honest, giving proper care to birds is MUCH more involved and time consuming than dogs or cats...IDK about rats as I've never owned one.
It's great that you are looking in to bird ownership and asking questions before jumping in to it tho. Good for you!!
I think at this time you should focus your time and energy on vet school..and of course the animals you already have need your time too.