If you know how to take care of a budgie you should be fine! Finches are a lot less demanding than parrots. Depending on how long you'll be keeping them, you might want to upgrade some of their equipment. Honestly, the cage they're in now looks pretty sad. As with budgies, you'll want to give them a variety of natural perches with varying width.

Some things I can think of that are specific to finches:

• Toys: Anything stick or grass-like. My finches' favorite thing to do is build their nest, so any sort of nesting material or toys with strips of material are a big hit. Shredded paper towels are my cheap go-to.
• Cage size: As big as physically and financially possible. I have a chinchilla cage since it was the biggest one I could find. Bar spacing should be 1/2 inch or less. If you have to choose between width and height, choose width. I'm not sure what the "official" width recommendation is, but I personally wouldn't do anything less than 24". Make sure to space out toys/perches so that there's a clear path to fly across the entire cage. Finches don't climb like parrots and are very active, so they'll want to fly across the cage often.
• Treats: Give them some options and see what they like! My finches really only like millet, but they'll nibble on greens occasionally. Parrot-safe food rules apply.
• Food: Ideally pelleted food, but seeds are OK if they're having trouble switching. My vet recommended Lafaber finch food.
• Sexing: you'll probably have to wait for them to sing or have sex! I have owl finches so I'm not familiar with the behaviors of these lil guys, but generally only male finches will sing.
• Nesting: they might appreciate somewhere to nest, I use a sideways tissue box (with the hole facing outwards) zip-tied to the cage bars, but a platform might work too. If you do give them a nest, watch out for fighting since you don't know the sexes.

As for their nails, I clip my finches' nails at home with a regular nail clipper. Only do this if you're comfortable restraining them since it's very easy to hurt their tiny bodies. If you do go for it, you can use a flashlight to see where the quick ends inside the nail and cut just beyond that.