Ok, so you want finches but don't want to deal with the breeding aspect of finch keeping. Totally understandable. Breeding finches can be very stressful at times and you never know when something may go wrong.
The first thing you need to do to make sure your finches don't breed, is to remove all nests and nesting material from the cage, flight or aviary. Without a nest they'll have a very hard time incubating the eggs the females will lay. I'm not saying they won't try with every fiber of their being. Reproduction is the #1 goal in all living things, it's how we survive.
You'll probably find makeshift nesting locations on the floor, in the food or treat cups, and in any plants you may have within their reach. When you do find these, don't worry. All you need to do is remove all the eggs you find. It's not a bad idea to check all the locations once a week or when you clean their cage. If the eggs are cold, it means they aren't being incubated properly and will never hatch. If they are warm, you'll need remove that nesting area or check it more often.
I know, this is sounding like a lot of work and you don't want the extra work. You only want the pretty finches. So we'll move to plan B. Single sex aviary. Everyone is a female or everyone is a male. Now there is no way they can produce young. An all female aviary will still produce eggs. Many of them in fact, but you can remove them once every few weeks if you like. No chance of them being fertile so there is no need to hurry with extraction.
Same sex aviaries aren't without their share of problems though. You really need to know about the species that you plan to buy before you bring them home. In some species, Orange Weavers for example, 2 males can not live together in peace for long. Eventually one will kill the other. Some species of Waxbills are the same way and even Gouldians have been known to show aggression in same sex aviaries; both sexes. It's not always the males who fight. Females can be equally as aggressive.
The trick to making a same sex aviary work, or at least to give it a better chance of working; is to buy the finches young. You don't want them too young however, because they may not handle the move well. A good time is when they are finishing up their first molt into adult feathers is a good time. You'll also need to introduce them to your aviary at the same time. Adding a few here and a few there may lead to territorial disputes.
The small hitch with keeping finches of the same sex is being able to tell the sexes apart. Zebras, Gouldians, Cordon Bleus are no problem. The two sexes look very different.
Species that can cause problems are Weavers and Strawberry for example. The two sexes look very different in the breeding season but when not in breeding season they are identical. It would be easy to pick up a group of females only to have a few of them molt into male colors several months later. In the end at least you can figure out their sex.
Others like Waxbills, Owls, and Society never change color and the sexes always look the same. Not all males will even sing if there is a more dominant male in the aviary. You buy a group thinking they are all males or females only to find eggs and possibly babies one day. No idea who the parents are.
Now you've got problems. Do you get rid of all of them and try again, or remove all nests and go back to the first plan and remove all eggs as you find them. So many decisions! Nothing is simple. The finch loves to surprise you and will. :)