Finches are very small birds that need to be handled very carefully. It's important to handle them with such care so that you can prevent hurting or damaging their wings or other parts.
The best way to handle them is to approach them from behind and as you gently grab the bird, make sure that you wrap both sets of feathers inside of your hold. Finches have a small wing span which will enable the bird to escape your hold very easily. Do not squeeze your finch but hold your hand in a firm but cupped position to prevent harming the bird. If your pet finch happens to escape then you might have to result in using a small net to catch them. You might have to chase them around on the floor with the net for a bit but you can catch them with it effectively. The net also provides for a greater method of capturing them unharmed. Once you have captured them in the net, place your hand over the net, hold it tight and then you can transport them back into the cage.
Also, you will want to make sure that you are not smothering the bird or preventing it from getting air. As you hold them make sure that their neck and head are exposed from your hold.
It will help to calm the bird if you are not screaming or getting excited as you try to capture the bird. In fact getting excited or shouting while trying to catch the bird will make the bird all the more eager to want to escape from you. Talk to the bird in a calm and reassuring tone.
The more you expect your finch to being handled by you each day it will no longer try to escape from you (as much). Usually a pet finch will hop and perch onto your hand when you slowly move your hand towards the bird.