Simple Climate Models that Predict Climate Change
It sounds like you are asking for a simple climate model that treats the globe as a homogeneous system.  A Zero-dimensional energy balance model may be of interest (see this Penn State site ) which uses a starting point that is also basically described in this post: Temperature as a function of luminosity and greenhouse gas concentrations 

If you solve for the temperature of the Earth using a blackbody approximation and thermal equilibrium (with an albedo of about 0.3), you are only taking into account the temperature of the Earth without a greenhouse effect (about 33 °C cooler than actual temperature).  It is very difficult to calculate the influence of infrared radiative forcing components (e.g. oxygen and carbon dioxide) to an empty atmosphere, since there are many biogeochemical/physical feedbacks in the climate system which are nonlinear.  

However, if you start with current temperature/concentrations, you can add terms that represent particular increases to the GHG concentrations.  Those terms are constantly being refined by the scientific community and are summarized in this Wikipedia article about Radiative Forcing (see the link to climate sensitivity at the bottom of the page).  This is a kind of "back of the envelope" modeling approach that most climate scientists use to discuss past and future influence of climate forcers, which can be very helpful when discussing the expected climate effects of various emissions/mitigation scenarios.

The Radiative Forcing article in Wikipedia says:


  Radiative forcing can be used to estimate a subsequent change in equilibrium surface temperature (ΔTs) arising from that radiative forcing via the equation:


$$\Delta T_s =~ \lambda~\Delta F$$


  where λ is the climate sensitivity, usually with units in K/(W/m2), and ΔF is the radiative forcing. A typical value of λ for CO2 is 0.8 K/(W/m2), which gives a warming of 3K for a doubling of CO2 [from pre-industrial times, assuming RF of CO2 is ~ 3.7)].


It's important to note that most climate sensitivity parameters and radiative forcing values are derived with respect to some time period (e.g. 1750 - 2011) and GHG feedbacks are set for a particular climate regime.  Revised radiative forcing and climate sensitivity parameters are needed to make calculations for other climate regimes.  

The IPCC AR5 figure of Radiative Forcing components attributes 1.65 W/m2 to the actual increase in CO2 concentrations from 1750 to 2011 (or ~1.3K in temperature rise, if lambda is 0.8).



The values of Radiative Forcing that IPCC provides are likely close enough for doing realistic scenario testing.  However, something in an entirely different feedback regime would need very different parameters.  For instance, halving the CO2 in the atmosphere would have a much greater absolute effect on temperature than doubling it; temperature increases logarithmically as CO2 concentrations increase. This is because strong CO2 absorption bands are nearly saturated under present conditions, and increasing CO2 from present conditions mainly affects the flanks of the absorption bands.  However, water precipitates out of the system easily under low-temperature conditions, so CO2 has a much greater forcing value in those types of climate regimes. 

The most difficult problems with testing simple climate scenarios are related to the oceans and the biosphere, which could have countless effects on the average climate and GHG regime.