No. You are at the wrong level of abstraction. Let me give you an example to illustrate your error.
Lets say there are vectors in 2-dimensional euclidean space. They have a direction and a magnitude (defined in the usual way). Given a vector, how would you describe it? Let's agree that the coordinate system has an orthonormal basis (x & y axes are orthogonal and 1 unit in x direction has same magnitude as 1 unit in y direction). These are your common x,y Euclidean coordinates. Now someone says here is a vector V. It has coordinates (1,1) {which indicate the line from (0,0) to the end-point of the vector (hence give its direction and magnitude). Suddenly, (to be dramatic) someone else says:"What?! No, no the coordinates are (√2,0)!" Then, someone else says "Huh? They're (0,-√2)!" My question to you is: can all three of them be right?" The answer is: yes. How? They each chose different directions for their x,y axes. The point is that the vector exists INDEPENDENTLY of the coordinate system being used to describe it. Its DESCRIPTION (representation) depends on the choice of coordinates (frame of reference) but it (its existence) does NOT. So, you need to understand that if you allow different coordinate systems, then the coordinates of a vector (or any line, or point, or surface, or volume, etc.) depend on the coordinate basis so comparing (1,1) and (0,√2) can not distinguish whether they are the same vector or not UNLESS we know what the coordinate frame being used for each is.
Once you grasp that a point or a vector are expressed as coordinates, then it follows that a metric is also independent of the coordinate frame but its representation is NOT. So, unless the frame of reference is agreed upon, it is meaningless to talk about g11, etc. (because the components depend on the coordinate basis).
In a Riemannian Vector Space, length (magnitude) has certain characteristics which other types of vector spaces may not share. The metric is a way to define those characteristics so that changes in coordinates (which do NOT change the space) keep the properties of any vector invariant (unchanged). (Any change to the vector would NOT be just a change in coordinates, rather you actually create something different, while a change to just the coordinate system and the vector's coordinates keep the vector "the same" (just its representation changes)).
The metric is a tensor. A tensor is DEFINED by the way it behaves under differentiation (with respect to the two coordinate systems (the new one and the old one, say). In general there are two possibilities. For any set of indices, it can remain invariant when differentiated with respect to the new coordinates OR it can remain invariant when differentiated with respect to the old coordinates...(if it does neither, it is by definition NOT a tensor). For Special Relativity the indices run from 0 to 3 (usually...compared to 1 to 3 for 3-dimensional euclidean space). A metric is generally a set of one or more n x n matrices which describe the transformation from one coordinate system to another. For a vector, you only need one matrix. To transform objects more complicated than vectors (ie tensors of higher order) you need more than one matrix, meaning a matrix of matrices. Note that in the simplest case, there may be no difference between the contravariant and covariant transformation rules (differentiation wrt new or old coordinates).
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All this mumbo-jumbo aside, the best way to understand the metric tensor is to use it. You need to see it being used in the equations of relativity, and how it acts to keep the underlying physical objects (particles, waves, forces, energies, etc.) invariant under change of coordinates (observers). You will grow more comfortable with it as you apply it. It is a learning process. I don't think you can really understand it except by application. Practice.
