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 learing problems arise from the following considerations:
It is enlightening to read about "the aim" of the modern algebraic geometry, so I'm referring to: motivations behind the 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 sheafs (so the concept of the admissible functions) etc. But, despite of this, when one goes into the real construction of the new objects, all theorems, lemmas and propositions are missing of details (that are left to the reader). For example the verification that certain presheafs are sheaf, functorial properties of assignements between categories and details about limits/colimits constructions are often missing. Even if the student has a solid background in algebra and geometry, generally he has not the time or the capacities to complete all the statements. Practically a course in algebraic geometry implies that one must take many statements as acts of faith. I realize that writers and professors may have the same difficulties (expecially 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 understand the constructions. Finally, a book or a course characterized by explainations and by motivating as complete proofs is much more instructive than a book or a course which cover many advanced arguments IMHO.
In mathematics when two object are isomorphic, is a common practise to "identify" them. Practically if $A\cong B$ but $A$ has a simple description we write $A$ instead of $B$, but formally we are thinking at $B$. This procedure is used very often in algebraic geometry, but in some cases without explaining the isomorphisms and in other cases the two object in question are considered "really the same" even if this can provocate formal problems (look for example here). This "abuse of identifications" often make lose sight of the essence of what one is studying and once again the "stupid student",exhausted, tends to simply to memorize things. I point again that the problem is not the abstraction, but the fact that the excessive tendency to simplify notations, often leads to inconsistencies.
Is not given enough importance to the following: the process of successive generalizations, put in place by the great mathematicians during the history, which marked the birth of the 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 seems to be informal dissertations at the eyes of the student that is eager of formalism.
In your opinion, what are the most common difficulties that a student encounters during his learning process of algebraic geometry? If my problems do arise precisely from the above considerations, can you give me some advice to solve them?