Many students - myself included - have a lot of problems in learning scheme theory. I don't think that the obstacle is the extreme abstraction of the subject, on the contrary, this is really the strong point of modern algebraic geometry. I'm reading many books such those written by Hartshorne, Gortz & Wedhorn, Liu, Vakil (notes), Gathmann (notes), Shafarevich, Perrin and Milne (notes) and in my humble opinion the learning problems arise from the following considerations:
It is enlightening to read about "the aim" of modern algebraic geometry, so I'm referring to: motivations behind schemes, the correpondence between algebraic and geometric entities (so the duality between the category of affine schemes and the category of rings), the importance of sheaves (so the concept of admissible functions) etc. But, despite this, when one goes into the actual construction of the new objects, all theorems, lemmas and propositions are missing details (that are left to the reader). For example the verification that certain presheaves are sheaves, functorial properties of assignments between categories and details about limit/colimit constructions are often missing. Even if the student has a solid background in algebra and geometry, generally they don't have the time or the capacity to complete all the statements. Basically taking a course in algebraic geometry implies that one must take many statements as acts of faith. I realize that authors and professors may have the same difficulties (especially lack of time) in writing down all the boring details, and moreover that a book with all proofs may include thousands of pages, but in this way students are encouraged (read discouraged) to simply memorize the most important results without really understanding the constructions. Finally, a book or a course characterized by explanations and by motivating as complete proofs is much more instructive than a book or a course which covers many advanced arguments IMHO.
In mathematics when two object are isomorphic, it is a common practise to "identify" them. Basically if $A\cong B$ but $A$ has a simple description we write $A$ instead of $B$, but formally we are thinking of $B$. This procedure is used very often in algebraic geometry, but in some cases without explaining the isomorphisms and in other cases the two objects in question are considered "really the same" even if this can provoke formal problems (look for example here). This "abuse of identifications" often makes one lose sight of the essence of what one is studying and once again the "stupid student", exhausted, tends to simply memorize things. I point again that the problem is not the abstraction, but the fact that the excessive tendency to simplify notation, often leads to inconsistencies.
Enough importance is not given to the following: the process of successive generalizations, put in place by the great mathematicians across history, which marked the birth of modern algebraic geometry. This process is fundamental in learning because it probably represents the most natural way whereby the human mind can deal with the subject.
In summary, because of the above issues (principally the first two), rigorous mathematical statements, incredibly seem to be informal dissertations at the eyes of the student that is eager for formalism.
In your opinion, what are the most common difficulties that a student encounters during their learning process of algebraic geometry? If my problems do arise precisely from the above considerations, can you give me some advice to solve them?