There are two ways to house finches Cage or Aviary.

All finches need a larger cage size. The larger you can give them the better.

They also do very well in aviaries, which may be outdoors as long as an out-of-the-weather and out-of-the-draft area is provided.

For those of us without the benefit of an aviary, a large flight cage provides adequate amount of exercise.
The width of the cage is more important than height and depth.  At least 30 inches in width is needed.
Minimum size for one pair of finch should be 30″x18″x18″ preferably 60″ wide if at all possible; the larger the better.
A finch cage is a sturdy indoor home. It keeps him safe from escape and household dangers while allowing you to view him and enjoy his activities.
Height is not as critical as having proper length and width.

Breeding cages can be smaller as long as the birds are allowed in the larger flights during non-breeding times of the year.

I have found finches like owl finches and parrot finches really breed a lot better in larger flights as they are very active birds.

Many birds can slowly acclimate to both higher and lower temperatures, but sudden temperature change can be very stressful and induce illness.
So when getting new birds, be sure to know what temperatures the birds are used to and allow them to adjust slowly to temperatures at your location.

Also a responsible bird owner, you should never allow your finch the run of the house given his small size, nervous disposition, and swift flying abilities.
Therefore, you will have to house him. There are two methods for housing finches: the cage and the aviary.

Cage Materials
Most cages are made of metal or some combination of metal and plastic.
The best cages are made of either wrought iron or powder-coated steel. These materials are extremely durable and completely safe for finches

If there is too much space between the bars of the cage, your bird may be able to get his head, leg, or wing caught in the bars. As a rule of thumb, the bars should be no more than 1/2 inch (1.5 cm) apart.

The closer your finch’s cage is to a basic cube or rectangle, the better. Round cages and cages that have fancy metal scrollwork may look nice, but they are less than ideal for your bird. In these cages, the bars will taper in some areas, which can result in your finch getting his head or foot caught.

There are a number of other features to consider when deciding on a cage.
One is a pull-out bottom tray. This will help speed up cleaning time considerably, and most modern cages will have one.

Be aware that some finches are small enough to fit through the opening when the tray is pulled out for cleaning.

If your cage doesn’t come with a seed catcher, plan to buy one. These drastically reduce the amount of food that ends up on the floor.

The Aviary

Think of an aviary as a walk-in cage that houses several pairs of finches. It will need to have all the basic features of a cage (perches, food bowls, etc.) but on a larger scale. An aviary can be indoors or outdoors or both.
While there are some special considerations for the outdoor aviary, the basic principles are the same whether it is located indoors or outdoors.

Materials
You can buy or build an aviary. It should be sturdily constructed of a wood or metal frame with a fine wire mesh. If the aviary is outdoors, concrete makes the best material for the floor. Make sure that the floor of an outdoor aviary slopes slightly for better drainage.

Predator-Proofing
If the aviary is outdoors, you must consider the predators that might live in your locale.

These include hawks, rats, cats and other animals.
Different predators require somewhat different methods to keep them out.
A good general rule is to have two-ply mesh walls with a few inches (5 cm or so) between each ply.
If you are using a concrete floor, have a metal or plastic skirt around the bottom of the mesh walls.
Frequently inspect the aviary for any signs of wear or intrusion.

Shelters
The shelter is a place where your finches can go to sleep in at night and escape inclement weather. It can be simple, but it must be solid and sturdy—most shelters are made of wood. Place your shelter in an area of the aviary that will stay comfortable for your finches no matter what the weather is like. This may require that the shelter be heated or have mesh windows for ventilation. It is very handy if the shelter can easily be closed off from the rest of the aviary.

Cage and Aviary Furnishings

Perches
Perches can be made of many different materials, including wood, plastic, concrete, and rope. Plastic perches are not a good choice, but the others are all fine. Most finch keepers use a mix of different perches. The only caution is to use only one concrete perch (sometimes called a sand or cement perch) because this type can be hard on your bird’s feet. If you use rope perches, inspect them regularly and trim off any frayed strands before they become long enough to entangle your finch.

Do not buy sandpaper covers or other perch covers. These harbor bacteria and can cause foot infections.

Bowls
You will need a minimum of three bowls for feeding and watering your finch: one for water, one for seeds and other dry foods, and one for fresh foods. The best bowls are made of stainless steel and bolt to the bars of the cage. Heavy-duty ceramic bowls are also good. Plastic bowls can harbor bacteria and are sometimes easy for your bird to destroy. It’s a good idea to have an extra set handy so that you can switch bowls in and out easily at feeding time.

Baths
Most finches love to bathe espically parrot finches, so consider purchasing a birdbath, or you can use any clean shallow dish.

In an aviary, you can have some permanent water feature. However, you must clean it frequently or it will become a source of infection for your finches.

The benefits of a water feature are access to bathing whenever your finches want to use it.
Dr Rob Marshall KD Powder can help keep water sources clean and also clean your birds feathers when they bathe.