I have recently started to develop my mathematical intuition. In the past I saw math as a mere game of symbol manipulation, whosoever was able to see patterns and cram formulas and apply them upon those symbols won. However,I now feel that the truth is contrary to the aforementioned,this is because I see in each new chapter that I study in calculus or linear algebra, there is an inherent geometrical intuition that can be easily visualized. At other instances there may not be a geometric interpretation, however there are ways of understanding why the proved results are true. But there is still a void of intuition in the most basic algebra that I am encountering. The following link should demonstrate what algebra I am talking about: http://link.springer.com/chapter/10.1007/978-0-8176-4549-6_1#page-2
Another example further in this text: Finding the roots of the equation: (xy − 7)^2 = x^2 + y^2
Although, the aforementioned can be easily solved after looking at solved examples, however, for each step my motivation is to basically juggle the symbols until I find a pattern I am looking for. What that pattern represents and the symbol juggling itself seems like an opaque layer to me? Please, can anybody give me suggestion to how I can intuitively plan out my steps doing basic algebra. Is there a way of doing math on an intuitive basis from the bottom up? Preferably, in a visual way?
This is not really an answer, but it was getting too long to be a comment.
Mathematics draws much of its power from deep, sometimes mysterious dualities between geometry and algebra, so I do not think there is any way to understand the relationship between geometric intuition and symbol manipulation in general.
Diophantine equations are one of the least geometrically intuitive areas of mathematics, though it is certainly worth mentioning that geometric insights (albeit very difficult ones) are behind some of the deepest results in the field, like Fermat's Last Theorem.
One can sometimes make arguments from intuitive principles that give partial answers. To take your example of $(xy-7)^2 = x^2 + y^2$, one can observe immediately that this is the intersection in the plane of two curves, one of degree $2$ and one of degree $4$, that have no components in common. By Bézout's theorem, there are at most $8$ intersection points (in fact, we can conclude that there are exactly $8$ solutions, if we know how to count them).
This tells us that the integer solutions, in particular, are a finite, possibly empty set. But it doesn't really give a good a priori method for finding them. That requires some symbol pushing.
To some extent, we can gain a lot of intuition for pushing symbols around. But this can be a lifelong process, and I think any mathematician will tell you that there are advantages to being able to manipulate equations without completely understanding what you are doing. There is some truth to the famous quotation that mathematics is about getting used to things, and for myself I can say that diligent practice has been far more valuable than theory when it comes to developing comfort with difficult problems.
The theory and intuition comes eventually, and with specific examples you may get little boosts to your insight here and there from helpful mentors, but there is no singular approach that will allow you to "plan".