Choosing approximate eigenfunctions In the variational approach of estimating the ground state energy of a system we choose an approximate eigenfunction dependent on certain parameters and then minimize the expectation value of the energy with respect to these parameters.
Is there a general approach to selecting these approximate eigenfunctions? 
Also if we do not know the actual energy in the first place how can we incur that the approximate eigenvalue is close to the actual value?
 A: In selecting a trial solution, the tests cases you try might be facilitated by a two step process. 
Step 1:
Figure out what standard problem matches yours PHYSICALLY. For example, is your system bounded? Is the bound complete like in the case of the infinite potential well in which nothing can leave a designated region? What is the symmetry of your system? In these cases you might want to base trial solutions off of The infinite square well, or the infinite spherical well, depending on geometry and symmetry. 
If you are dealing with oscillations, you probably want some variation of solutions to the QM Simple Harmonic Oscillator. If you have an inverse square potential, you want to look at trial solutions from the Hydrogen atom. 
Keep in mind the Parity of your model. You might want to restrict your trial solutions to  even or odd functions from your base model. 
Step 2:
Now that you have selected which Physical basis to used and determined your geometric and  symmetry/parity requirements, you can consider simplified MATHEMATICAL versions of those solutions. Typically, you want to replace the solutions with a decent polynomial approximation. 
For example, the ground state of the Infinite square well is 0 at the ends, peaks  in the middle and has even symmetry about its line of symmetry. y=Ax(L/2-x) does exactly this while only being a polynomial function. 
In the case of the Infinite Spherical Well, you'll need some Mathematical approximation to the early Spherical Bessel functions. I don't remember off hand, but there are some standard polynomial approximations for these. 
If you are dealing with any system that oscillates, you can get close to a ground state solution by selecting trial functions from the solutions to the Simple Harmonic Oscillator. To do this, first you want to find your "spring constant" which is roughly the second derivative of your potential with respect to space coordinates, evaluated near the equilibrium point. So your trial candidates are the Hermite polynomials times $e^{-px^2}$ for some parameter p representing your base energy. 
Energy Concerns:
You usually know something about the energy of your system from the potential. The total energy cannot be smaller than the minimum of your potential. If your energy is much higher than max of your potential, say five to ten times more,  you hardly have any potential at all, which allows you to approximate the potential with a Trial Function before applying further analysis. Your "spring constant" is proportional to the second derivative of your potential where the first derivative is zero, call it $k$. 
If you have a characteristic length scale, say the radius of your sphere of interest, $r_0$, a useful energy might be on the order of $\frac{1}{2}kr_0^2$.
By the Uncertainty Principle you can associate with a characteristic length, a momentum of $p_0=\hbar/l$ corresponding roughly to energy of $p_0^2/2m$. This approximation is incredibly accurate for the ground state of the harmonic oscillator. 
