Do you need time or a mentor to train new force powers, or can a character just learn it without needing an in-game justification? (Thus for move-basic if you haven't learned it yet.)
The basic response I would give is that it depends on the style and length of the campaign you intend to play. What does the group want their Force users to be able to do during the bulk of the campaign, and how quickly should they reach that point? The game itself can support a very loose or a very strict style of play.
In my campaign game, for example, I work in concert with the players to assess what is needed to justify each increase in a Trait. We accept that in-game justifications are appropriate for the type and length of play we want for these characters. Sometimes, it makes sense for a trait to improve 'on its own.' Most of the time, it does not. In my recurring one-shot game, however, improvement from episode to episode of play is desirable, so there are no restrictions placed on XP spending other than 1 level increase per trait. It is related to the style of play desired.
Rules for Improving Traits are found in Edge of the Empire on pages 29, 30 and 92. In Age of Rebellion this information is found on page 36, 102, and 317. In Force & Destiny, they are on page 37, 102, 103, and 320. The rules cover buying traits in character generation and afterward. The section entitled Awarding and Spending Experience Points in Chapter 9 of each core book makes reference to the material given in earlier chapters, so it can handle questions of basic XP use.
"Training, you need!"
The Force section in EotE and AoR both contend that the Force Sensitive Exile and the Force Sensitive Emergent templates are the extent of Force Ability which can reasonably be learned without a teacher. While it does not provide a firm, "No, you may not improve beyond this without a teacher edict," it does state:
"Most Force sensitives would not be able to achieve the degree of skill necessary to become a fully fledged Jedi Knight, and in the modern age the Jedi have been so thoroughly expunged that finding a qualified teacher in their techniques is a practical impossibility."
Age of Rebellion, Chapter 8: The Force, page 291
It does not put a firm or specific freedom to learn the powers in EotE and AoR without a teacher either. The game is written in a manner which supports and approves of creative control by each group. In contrast, a relationship with a mentor, or access to training materials such as a holocron, ancient temple, or the like is given more emphasis in Force & Destiny. As the latter game is specifically about the experience of being a force-sensitive, this makes good sense.
The Fantasy Flight Games Star Wars RPG takes a very narrative stance on many elements of game play, including how experience is spent. This is mirrored across the three main extensions of the line (Edge of the Empire, Age of Rebellion, and Force & Destiny).
A hard line on training, time, and mentors is not clearly spelled out for any Career, Skill, Specialization, Talent, or Force Ability (yet), but the implication may be felt as one reads through the material. This is being left in the hands of the play group to determine what is and is not appropriate for development at each specific juncture where experience is spent.
One could gather from the text that as awards are to be doled out after each session in increments of 10 or so XP, that players might be free to spend these XP on their characters' traits at any point. There are some situations where this makes sense, such as a story line which involves a lot of repeated uses of certain skills over an extended period of time. There are others where it does not, like an action-packed set of sessions involving a lot of conflict, but relatively little time. The text does indicate, however, that periods of downtime should be given for characters to learn new Skills and Talents.
Assessing if enough time and specific experience has been gained to make an improvement make sense within the ongoing campaign. The implication of in-game justifications for improvement is present, but not stressed as a rule, leaving the group free to adjust as needed, or to suit a specific situation.
It may make good sense to require XP awards to be held until specifically designated improvement points, wherein time can be allotted to cover aspects of mundane life such as training, networking, and development.
As descriptions of the source of XP awards are recommended to be shared, by both sides of the screen, it could be understood that the idea of relevant experience is an aspect of the game. What have the characters been doing to warrant subsequent improvement. Does this practice or development occur solely in downtime with a trainer, mentor, or information source, and/or are they actively demonstrating training in play? These things are shown in the source material, and implied in the core book, but again - not set into a rule. The group is again, free to adjust as necessary to suit the ongoing tale.
Assessing how the character learned to improve a specific trait, Force or otherwise, may be good for supporting nuanced play, which generates more hooks and potential paths to explore as a result of the requirement to have in-game justifications for improvement.
Speed of Development
Characters who have received in the neighborhood of 200 XP are quite accomplished and while they still have gaps in their knowledge, are extremely capable in their core proficiencies. They are not to be trifled with, and can overcome significant obstacles. Simply spending XP as it arrives may push the group into a rate of progression it does not enjoy, or which is not sustainable for the long-term.
Again, setting a habit of providing in-game justification for development can allow characters to build a pool of XP which the player is then able to spend as need and opportunity arises, rather than seeking places to stick spare points.
Force Abilities as presented in Edge of the Empire and Age of Rebellion are potent, and the Talents which accompany the Force User Templates add to that. That said, the Templates and Abilities shown in Force & Destiny demonstrate that much more potency awaits. Rate of progression may still become an issue in a group, but if the character is restricted to the Abilities provided in EoTE or AoR, potential concerns about overall power for the character can be lessened. The Force specialists in F&D will still seem very special and different, regardless.
How are these abilities learned? Again, the implication would seem to be from a combination of things which make sense in the ongoing campaign. Trial and error, trusting one's feelings, finding information such as Holocron Cubes, or bitter Force Users with a bit more knowledge and a willingness or a weakness which will convince them to share. Again, by following the opportunity to limit development to specific training sources and opportunities in the game, richness of play can be bolstered. Rate of development can also follow a speed the group finds comfortable.
In contrast, a group slated to play for a short time may desire a faster rate of development with the need to seek a teacher or source of insight hand-waved into the background.
Experimentation may be required or desired. In my first campaign of Edge of the Empire, our Force User in Exile was allowed to develop Move through trial and error to a fairly capable level. The powers manifested after our first 'story' ended. He chose to improve it after each 'story break for XP use' thereafter. Seeing how this rate of expenditure affected his overall competency with that power, guided me in assessing how his second power would progress, and I slowly applied in-game reasons to tie that progression to other sources of development than mere trial and error.
I found that this approach mirrored what we see Luke go through in the films, and after his time with Yoda, we can imagine a drift back toward a greater need for trial and error, but also a greater capacity to receive tutelage from his ghostly allies. For characters without help from 'the living Force,' a reliance on sharing progress with one's friends, allies, and mentors will help a long-term group deepen the experience of play, and control the rate at which characters improve in order to maximize the time spent in whatever sweet spot they perceive and enjoy in the system. The game supports this through strong implication, but leaves room for other approaches.
In Force and Destiny, having a Mentor will reduce the cost of buying a new force power, but not any of the enhancements to that power. Having a Holocron may allow you to learn certain skills that the GM otherwise would not let you learn, and should give you access to one or two additional skills as Career Skills, so long as you have that Holocron.
In the EU, there are examples of people going off to research, study, and otherwise learn new powers, but without any mentor to help them.
So, given the narrative aspect of the FFG Star Wars Role Playing Games, the most important thing is what the GM wants to allow in their game, and how they want to allow that to unfold.
Let me dig out my books so that I can hang some reference page numbers and quotes on this answer.