In the wild, if a tortoise is picked up, it is usually in trouble! A bird of prey has picked it up and is going to drop it to break the shell open, or a big cat has grabbed it and plans on breaking off the edges of the shell with its powerful jaws until it gets to the meat, or something else bad is happening.
When we pick up a tortoise, its first instinct is to get away, so it thrashes and claws at whatever it can. Another option is to pull in tight- which limits its ability to breathe. Sometimes they get so stressed they can go limp- this is often a pretty bad sign of a weakened, stressed animal.
There are three schools of thought about handling exotic animals that boil down to ‘eave them alone as much as possible’ one one end, ‘socialize them a lot’ on the other, and ‘get them used to being handled but don’t overdo it’ in the middle.There are pros and cons to each group, and all three groups have had successes in their areas- tortoises that show perfectly natural hiding behaviors all the way to tortoises that act almost like dogs in showing what seems to be affection.
First- the safe way to pick a tortoise up is to just pick up the shell however you can do so safely, then support the bottom and limbs as much as you can. A tortoise being held by the shell struggles, a tortoise sitting on your hand is often calm and looking around. Make sure you keep a hand on them at all times, though- we don’t want them to walk or slip off and fall! As a tortoise gets larger, it gets harder to support the limbs, and some larger tortoises may need two people to lift them safely.
Getting a tortoise used to being handled has some benefits- it reduces the stress for the times you have to handle them- cleaning the habitat, medical treatment, etc. To get them used to this, just pick them up and hold them. For the first couple weeks, do it once or twice a day for just a minute or two each time, then set it right back down in the habitat. Offering a small treat will help them associate this with something good. As they get more comfortable, extend the time they are being held until you can hold them for several minutes with minimal struggling.
If they start to pull in tight, they are getting stressed again, so put them back down somewhere safe.
Many tortoises appreciate being rubbed on the head or stroked under the chin, or having their shells petted (but not tapped), Cooing or making other noises may relax you but they cannot hear most human sounds- although they can feel the vibrations.
Tortoises are smart, can see well and in color, and have a good sense of smell, so it does not take long for them to figure out that this one specific giant creature gives them food, and build a relationship with the caregiver. If you give them a cue just before you feed them (hit the wall of the outdoor pen, knock the table the indoor tub sits on, etc.), they will quickly learn to associate that cue with food also.
Tortoises can be trained, but it is a bit trickier to find a good reward and incentive, and they are a lot less interested in pleasing their trainers. In that sense, they are more like cats than dogs. Small bits of fresh or dried fruit or meat work for most species (emphasis on small), and a stomp on the ground or stroking the shell can replace the clicker or verbal reward.