i bought some niger seed and the goldfinches didn't touch it. all i use now is sunflower hearts, i find the black sunflower seed husks which the birds don't eat make such a mess on the ground, same with the seed mix stuff. i get all sorts of birds at the sunflower hearts, including loads of goldfinches who love the sunflower hearts. if they're in your area, they'll appear. we also have 3 bullfinches who regularly visit the feeders which i'm delighted about as one died in the garden last year, along with a whole load of disappearing greenfinches, due to the virus which was rife. thankfully the greenfinches are also back but not in the huge numbers they were.
i also have a couple of peanut feeders, the mesh sort for whole peanuts, and during winter i put out fatballs and mealworms. i also buy smartprice sultanas and throw a couple of handfulls of them around under the feeders. if you have any apples in the house which are going soft and you won't eat, cut them in half and put them out, the blackbirds love them. again, during the winter i buy smartprice apples especially for the birds. i've put blackening bananas out too, which get a mixed reception but generally get eaten in the bad weather.
personally, i'm prepared to take the risk of a rat visiting and always put food on the ground for the ground feeding birds.
the rspb site is the best one for advice.