The Jedi
There is no emotion; there is peace.
There is no ignorance; there is knowledge.
There is no passion; there is serenity.
There is no chaos; there is harmony.
There is no death; there is the Force.— The Jedi Code
"Emotions are a major source of disharmony within ourselves," she told him. "They are a natural part of our existence, but too often sentient beings allow their emotions to control them, causing them to act irrationally. And so, we have the Code. There is no emotion; there is peace. We counter aggression with stillness, chaos with tranquility.
"Jedi do not abandon their emotions, Josh," she said, encouragingly, reaching out to touch his arm, "but instead they learn to harness them. To understand them and use them in a positive, constructive manner." (source)
Powers
Well, there's healing (I mentioned the Jedi healing trance earlier). There's Force speed, which Qui-Gon and Obi-Wan used to run from the destroyer droids. Jedi can apparently hold their breath for extended periods (from the beginning of Episode 1, again).
Jedi have a form of precognition that could be called a danger sense; it's what enables them to use their lightsabers to block blaster fire.
They can enhance their senses (eg making their hearing sharp enough to hear a conversation across the room) and their short-term memory, recalling fine details they may have only passingly seen.
They can comprehend most alien languages.
I want to stop short of saying that they're telepathic. I don't think they can actually send words to each other. I think of it more as a form of empathy - impressions and feelings. So a Jedi could tell if you were hungry, or that you were worried about a friend, but couldn't find out what your credit card number was.
Other than that, there are a lot of telekinetic powers - leaping, slowing enemies down, throwing things, etc. I'm pretty sure that Jedi can't fly. I'm not sure if they can levitate themselves, but it seems plausible.
Jedi training
She would train her apprentices in the evenings at the apartment - together, to start with, as she introduced Josh to the history and mindset of the Order. She felt that the refresher course might do Carr some good too.
In the communal sessions, she covered a Jedi's awareness through the Force, meditation, the basics of Jedi healing techniques, and sensory enhancement. As their powers grew, her two apprentices also became able to understand most alien languages.
She also taught them how to sense people's intentions, alter their perceptions and affect their minds (eg "you don't need to see his identification").
As time wore on (four or five months passed) and her apprentices proved to have different aptitudes, she began teaching them separately. She began covering the more advanced techniques such as bodily enhancement (running at superhuman speeds or leaping great distances), combat awareness and the manipulation of physical objects (lifting and throwing). She acquired lightweight stun rods and began drilling Carr and Josh in lightsaber patterns. (source)
After the first practice duel between Carr and Josh, she started placing more emphasis on the philosophy behind why the Jedi acted the way they did: why they had to act from a place of emotional detachment; why they should never be the aggressor; how their actions affected the fabric of the galaxy around them.
In her training sessions with Josh, she started drilling him with the practice remote more often - a small repulsor-powered globe that hovered in the air and fired harmless stinger bolts. Attempting to rectify her earlier mistakes, she gave him limits in these practice sessions: at times he could only use his stun rod one-handed - sometimes in his right, sometimes in his left; sometimes she forced him to use a two-handed grip; sometimes she played deafeningly loud music (much to the irritation of the other tenants in the building) or blindfolded him. At other times she didn't let him use a stun rod at all, but made him levitate a small piece of hull plating to block the remote's attacks. As his training continued, she increased the difficulty by adding a second remote, and turning up the stinging effect of their blasts.
She hoped that these exercises would help Josh to broaden his awareness of his surroundings, by forcing him to keep track of more than one variable at once other than just his opponent. She ran the sessions with the remote longer and longer, hoping to work on his physical endurance as well.
Cerena also continued Carr's training with the remotes, but started placing less and less emphasis on his physical training as it became apparent that he was well-advanced in that area already. Instead, she began focusing more on the cerebral, internalised skills with him - healing, breathing control, detoxifying poisons, the Jedi hibernation trance, and so on. (source)
The masters at the Temple had always encouraged enlightenment through self-awareness, meditating and turning within in order to unravel the issues of one's own heart. As a youngling, Cerena had rarely had cause to ask any of the masters for help with her personal problems. (source)
Lightsabers
"I've been giving some thought to how we might be able to craft lightsaber crystals of our own. I know it's possible, but I still need to do some research.
"Besides," she said, in a lighter tone, "a Jedi's lightsaber is his or her symbol, a very personal item. I'm sure you and Josh wouldn't want to be running around waving antiques in the air! Your lightsaber is an extension of who you are. It's best that you and Josh construct your own, when the time comes." (source)
"A Jedi should not judge things by appearance," Cerena said. "A lightsaber is not made to appear intimidating or to inspire fear. A lightsaber is a tool and a symbol - elegant and utilitarian at the same time." (source)