There are several things to take into consideration when purchasing a cage for your bird.
Of course the size of the cage required is going to depend on the size of your bird. At the very minimum and only if your bird will be out of the cage for much of the day, he should be able to wildly flap his wings without touching the sides of the cage or any of the toys. A small bird does not do well in a small cage. Birds such as finches are tiny, but extremely active and need a cage large enough to do more than hop back & forth between perches. There should be at least two perches on two different levels and not directly above or below the other one. If your bird has a long tail, the perches should be far enough away from the bars so the tail does not touch. There is no maximum size. Hopefully your bird will have much more than the minimum.
The normal shape for a birdcage is rectangular. Parrot cages are often taller than they are long and Canary or Finch cages are normally longer than they are tall. Round cages are not recommended as many birds do not appear to feel as safe in this type of cage as they do in square cages. Sometimes you have a choice between a dome top cage, which has additional room on the inside of the cage, or a playtop cage, which has a play area on the top of the cage.
The best material to have a cage made out of is stainless steel, but this is also the most expensive type of cage. Powder coated cages are another good choice. Powder coating is a method of spraying dry powder paint over the metal cage and then baking it into a durable finish.
Before using a new cage, check it over carefully for any sharp edges or any places where a bird might get a beak or a toe caught. Make sure the bar spacing is appropriate for the type of bird you will be putting in it. Check how the door closes & locks. What is fine for a canary or finch will not be acceptable for a cockatoo. Make sure the bolts are not accessible to a bird that may learn how to undo them.
This cage will be home to your bird for many years and you want to make sure that you give him a good home. Get the largest cage that you can afford and that will fit into the area you have set aside for your companion. There is a good chance that the cage will cost more than the price of the bird.
One other option for your birds is if you can build cages yourself. Amazon.com has a book with details on accomplishing this and you can check it out at How to Build Everything You Need for Your Birds.