When handling frogs, you need to use extreme caution. Frogs are, as may be obvious, very fragile, and can easily be injured by jumping out of your hands, or by you squeezing them too hard.
Before you handle or even touch your frog, make sure you have rinsed your hands thoroughly. Frogs absorb practically everything through their skin. Some salts and oils from our hands can hurt frogs. Sunscreen, insect repellants, lotions, soaps, dirt, germs and other chemicals can irritate the frog's skin badly. Don't use soap before handling your frog. Just rinse your hands and leave them partially moist.
Always, always, always wash your hands with strong antibacterial soap after handling frogs. Frogs can transfer diseases to humans that can be fatal. However, don't let this scare you out of getting a pet froggy! Transfered diseases are not common and the diseases that can be transferred are uncommon, too. Just make it a rule to always rinse before and wash afterwards. If you're extra picky, wear disposable latex free, powder free gloves when handling your frogs. Rinse the gloves before handling the frog, dispose of them after, and then wash your hands. You can't be too careful.
If you know your frog is sick, avoid handling it entirely. You will have to transfer the frog to a different terrarium in order to clean the one it was in. If you are housing any other animals (reptiles or amphibians), do not handle them either, because illnesses between reptiles and amphibians can be contagious to other animals. Keep pets and young children away from the sick frog. Since you will have to transfer your frog if he's sick, there are a few options (besides picking him up with your own hands!).
Larger frogs are harder; mainly for one reason: they're big and usually harder to hold. You will need a paper plate in this case. Since most all paper plates have a raised edge, you'll need to cut it off so it's easier to scoop up the frog. Wearing gloves, gently push the plate as far under the sick frog as you can.
These two ideas are only for frogs that you know are sick. It's VERY stressful for frogs to be held, touched or handled at all.
Try not to handle frogs unless: 1) you are cleaning their terrarium, or, 2) they are sick and need to be transferred. To handle frogs, what I always do is, using my thumb, pointer finger, and middle finger, put your pointer finger and middle finger on one side of the frog and your thumb on the other side of it. Curl your fingers and GENTLY scoop up the frog, closer to the head than not. Do not press hard, because a little pressure to us is like an earthquake to the frog.
Frogs have very soft bellies that can be easily damaged. Frogs do have bones, but they're small and fragile and, like ours, can be broken. Just a little too much push from your finger might do a lot of damage. Be careful not to squeeze the frog's abdominal area.
Frogs panic when they are going airborne in your hands, and will probably struggle and try to hop away. This is why you hold the frog in this position for as little as possible. You should already have prepared a container or flat surface for the frog to be placed in. The container should be full of about 1/2 inch of water. Some frogs produce toxins while being held, but these toxins need to be replaced. Water does the trick.