Of the functions you list, two are $q$-series, one is an Eisenstein series
and one is the eta function. A general strategy in the form of a
computational process to prove or disprove modularity in such a generality
does not exist. The subject is simply too rich and elaborate to be amenable
to a canonical line of attack. However, useful principles exist, along
with a fascinating theory that underlies the identities used to prove the
modularity of at least the $q$-series you presented.
What do we know about modular forms for a specific
$\Gamma<\textrm{SL}(2,\mathbb{Z})$ and a specific height? First of all, they
are finite-dimensional vector spaces. We have explicit generators, such as
Eisenstein series, theta series etc. whose
modularity is proven by methods (that I will mention below) interesting in
their own right and generalizable. We have operators on these spaces that
are diagonalized by nice bases such as Hecke eigenfunctions. See
http://maths.dur.ac.uk/~dma0hg/ModForms.pdf for a good, free introduction. The point is that spaces of modular forms have a very rich and complex
structure, and membership in that elite club should never be taken
lightly.
Let me start with a few basic methods for showing modularity, leading up to a
suite of structural statements related to the modularity of your series.
The most basic method to prove the modularity of a function is to have it
exhibited as a sum over representatives of $\Gamma$ in the first place. Then
an application of an action of $\Gamma$ will simply permute the sum and
hopefully alter the summands by a factor of automorphy, as in the case of
Eisenstein series; of course we are not done: we need to prove the analytical
conditions involved in modularity, which are very crucial and non-trivial. Fortunately, if we can prove these conditions for a few functions, and then
show that more complicated ones are functionally related to those in a nice ways, growth
conditions will be much easier to establish.
If a function is presented as a Fourier series over $\mathbb{Z}$ (so we at least
know it is invariant under $z\to z+1$), a uniform method to detect modularity
is the (not necessarily straightforward!) application of the Poisson
summation formula. This is one of the ways
to prove modularity for the theta function, for instance. Although this tool
may seem somewhat ad hoc, it is not: the Poisson summation formula (and its
big adelic brother, about which you can read in any exposition of Tate's
thesis) is one instance of a trace formula, a highly structural statement
linking the geometry of orbits of a group such as $\Gamma$ acting on a suitable space
and certain algebraic representations of an overlying algebraic group (such
as $\textrm{SL}(2,\mathbb{R})$). This is also the first indication that certain
aspects of Lie groups may have something to do with modularity.
Once we have proven modularity, directly, by Poisson summation or other
Fourier/complex analytic method of some basic functions, the next step is to
see if we can write our functions 'in terms' of the previous ones. What 'in
terms' means can differ wildly from function to function; for instance,
although the $M_k$ are finite dimensional, it is hopeless to expect a cheap
proof that some series is modular by exhibiting it as a linear combination of
some Eisenstein series (usually it goes the other way round: we show a
function is modular and then we stand gaping at the results as
finite-dimensionality gives us all sorts of impossible-seeming equations for
the arithmetically significant Fourier coefficients of each side).
One notion of 'in terms' that is directly relevant to the functions you have
is via differential equations (of hypergeometric type for instance), leading to relationship
between various $q$-series and (modified) theta functions allowing to prove
modularity of the former via the latter. This is a tricky thing, combining
elementary combinatorial techniques illustrated, for example, in
http://www.math.uiuc.edu/~berndt/articles/q.pdf and the method you can find
in
http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/978-3-540-74119-0_1/fulltext.pdf in section 5.4. There are entire books (e.g. the survey
Parititions, q-series and modular forms) relevant to these
techniques, but they are heavy-going and I would not recommend them at this stage.
In the techniques above, heavy use is made of certain functional
equations involving series, products, continued
fractions, integral representations and more. These functional equations
sometimes give modularity immediately or assume a crucial role in the
process. Examples include the Jacobi triple product identity, the
Rogers-Ramanujan identities (which immediately give modularity for some $q$
-series) and more classical identities of that form.
It turns out that many of these functional equations have a common
origin: the representation theory of semisimple Lie algebras and related
algebraic objects (such as vertex operator algebras, Kac-moody algebras
etc.) and in particular, formulas for the characters of such representations
in terms of combinatorial data. If there is a unifying
principle that links modularity to series of the form your present above,
this is it. To illustrate this unifying principle, you should read about the Macdonald identities, e.g. at
http://en.wikipedia.org/wiki/Macdonald_identities.
A good book to read for the link between modularity and representation theory
in a casual manner is a book I am currently reading myself called Moonshine
beyond the Monster. It covers many (too many) topics, and although it can be
vague and somewhat inaccurate, it conveys the general ideas in with good prose and includes
sufficiently many references to authoritative material. Plus the modularity
and affine algebra interpretations sections are among the most well-written in the book.
