In the Galactic Republic before the war, a Jedi Initiate receives lessons in the Force and in lightsaber combat, among other subjects, and will eventually construct his/her own lightsaber. When an Initiate is deemed ready to go out into the real world to conduct missions and a Master takes him/her in, the Initiate becomes a Padawan. A Padawan who has completed the Jedi Trials or an appropriate equivalent achieves full Knighthood.
By movie's end, Rey has demonstrated the ability to do the following like a trained Jedi (meaning she can control her ability to do this intentionally):
- Use Mind Trick successfully, which as demonstrated by Ahsoka Tano during the Clone Wars takes substantial time and training even as a Padawan to pull off reliably.
- Survive a lightsaber fight, using the Force to guide her.
This suggests that Rey's lightsaber proficiency may be good enough "for the real world" and her Force proficiency, at least in some parts, can match that of an experienced Padawan at least. She certainly has learnt the foundation to executing Force powers at will and by intent. However, she completely lacks formal training in the ways of the Force, and everything she knows is based off of rumours, stories and layperson knowledge, applied over a matter of mere days at the longest.
Luke Skywalker is a precedent that one with particularly strong affinity for the Force can gain proficiency rapidly and to a significant extent even without sustained formal training. Therefore, I won't question if this is at all possible. However, how would Rey fare in a formal assessment of her abilities? Where in the line of progression, defined in the first paragraph, would the scorecard place Rey currently (broken down into different aspects if necessary)? Given Rey's talent for self-learning the ways of the Force, what benefits would formal training by a Jedi Master give, and what is only achievable through such formal training?