I always have a wide selection of tarantulas available, some are far more 'friendly' than others! The calmest species are generally those found around the Mexico, Chile, Peruvian areas. Any spiders that are referred to as Baboon spiders are usually pretty but certainly not for handling and the same can be said for most Asian tarantulas. I often tell people to look at their pet spiders in the same way as fish, they make a great display animal in a tank but are not for handling. Tarantulas are delicate animals and should be treated with care. However, that said, the following are some of the more friendly and calm tarantula species. I have kept tarantula spiders for over 35 years and have a few pet ones that I take along to my Ginny's Jungle events and regularly hold. They are Matilda and Lily (Mexican Red Knees), Hairy Harry (Brazilian Black) and Curly Sue (Curly Hair). I'm pleased to say that none of my pet spiders have ever bitten me (or anyone else!)
Mexican Red Knee (Brachypelma hamorii)
The original pet tarantula! Large, brightly coloured and docile. These spiders make good pets. They live for many years and have a pretty orange and black colouring with stripes on their legs. This is the tarantula that is often seen on television and in zoos and butterfly houses. This spider is now protected, importation of new individuals is not allowed, all Mexican Red Knee spiders are now captive bred and are therefore usually more expensive than other tarantulas.
Chile Rose (Grammostola rosea)
This tarantula was the second most popular pet spider but they have become far more scarce in the last few years. They were imported as adults in large numbers but I understand that their importation from Chile has now been banned. They live for many years and are generally docile. They are light brown or sometimes a rusty red colour and do not flick hairs from their abdomen as readily as a lot of other spiders tend to do.
Chile Flame (Homoeomma chilensis)
The Chile Flame is a small, slow growing species. They are very docile but just like the Chile Rose they are now very difficult to find in the pet trade.
Brazilian Black (Grammostola pulchra)
This species has become very popular of late. They are extremely 'friendly', docile, attractive and do not kick off hairs.
Curly Hair (Brachypelma albopilosa)
An attractive spider with long dark brown hairs that curl around the legs and body, they are generally docile and live for many years.
Pink Toe Bird-eater (Avicularia avicularia)
A very pretty black spider with pink tips to it's 'feet'. There are several species of Pink Toe Bird-eaters available, all are very hairy, tree dwelling and build strong, thick webs, they are fast moving, but are not aggressive.
Mexican Red Rump (Brachypelma vagans)
Another popular tarantula, large, docile and long lived. This spider is jet black with beautiful red hairs on the abdomen. They tend to sit out in the open and therefore make a good show spider.
The above information is based on my own personal opinions and observations. If you would like to add anything or feel that something is inaccurate please do let me know.