Birds are very active and the bigger that cage the better.  Your bird will need room to spread his wings and stretch.  Bar spacing is important, you don't want to find your finch with his head caught in between the bars of a cage with gaps that are to big for him. Below is a table with recommended spacing for the different breeds:

Make sure that your cage is free of sharp edges, is sturdy and if painted make sure that it was done with non-toxic paint. You will need to change the papers on a daily basis (check stools during this time to make sure they are not runny or loose) and give your cage a good thorough cleaning at least once weekly.  Plain water and bleach makes a great disinfectant and will not harm your bird if you rinse the cage well. 
Do not place your birds in direct sunlight, and keep him away from drafts. They love to be up high, making him able to see all that is going on.  He will enjoy being in company with you and your family so put him in an area that has moderate activity.  DO NOT keep him in the kitchen, there are many, many toxic items and fumes that are deadly for birds in this area and keeping him there may be fatal to your pet.
Covering the cage is optional, but it is a good idea to give your bird 12 hours of peace and quiet.  If you will have lights on later in the evening perhaps you should cover him to make it possible for him to sleep.  Take a peek after covering him to make sure he isn't afraid, and if not keep him covered.  If it is an area that is peaceful and dark in the evenings, cage covers are not necessary.

Birds like to climb so have his perches set up so he can climb up and down his cage.  Use a variety of textures and shapes, he will love it and this is also beneficial to his feet and nails.   Concrete perches are especially good for this.  Putting a perch in front of his food, and at the door to his cage would be perfect and make it easier for him to eat and "step out".  A 1/2 inch perch is good for smaller birds, and for larger birds, such as the parrot, a 1 inch perch is perfect. The most important thing is that the perch fit the birds' feet.

Different sized birds need different sized toys.  They love toys made of yarn, wood, shiny and bright objects, mirrors.. the list is large and varied.  When you are looking for toys, the package will probably let you know what size breed it is for.  Your bird may be a bit picky and may not like what you choose, but be patient, eventually you will find something that he likes.  I've heard that some birds really like baby toys and other things around the house.  If you see something around that is sturdy and looks fun to play with, give it a try and see how it goes!