Effective Temperature of a Planet and Surface Temperature due to Greenhouse Effect
I think you’re not far off with your understanding, but maybe I can put in a few comments to make things clearer.


  I often get confused because most diagrams for the Earth energy budget and net radiation balance show more infrared leaving the surface than is ever supplied by solar radiation at the top of the atmosphere in the first place… As you can see, more infrared is leaving Earth than enters as solar radiation and yet this is still a picture of equilibrium where the net radiation is zero.


These types of diagrams only show the fluxes, but you have to remember that the surface (mainly the ocean) and the atmosphere are reservoirs of energy that have built up over time.  In that diagram the top-of-atmosphere net radiation balance for the Earth system is,

67 (absorbed by atmosphere) + 168 (absorbed by surface) = 235 (outgoing longwave radiation)

and all the other much larger components are just different pathways for energy transfer between those two internal reservoirs.  The sun only has to top-up the amount of energy that the reservoirs are leaking to space (235 W/m2) to maintain the equilibrium.


  This re-emission of infrared radiation to the ground is like a trap for energy and over time it will cause the surface temperature of the planet to rise… If the terrestrial surface is warming it will have to emit even more infrared radiation back to space... I was thinking that the surface will grow warmer and emit more radiation until the amount exiting equals the amount incoming, establishing the new thermal equilibrium.


Broadly yes, but remember that the diagram you posted describes a long-term steady state (“thermal equilibrium”), so all that “warming” has already occurred and is built into those numbers.  It may just be your phrasing, but note that surface doesn’t emit the bulk of its thermal radiation directly back to space.  In your diagram this is the 40 W/m2 “atmospheric window”, which is much smaller than the 350 W/m2 of surface emission that’s absorbed by the atmosphere.

It might be better to think of the atmosphere less as trapping energy at the surface and more as slowing the net transfer of energy from the surface through the atmosphere to space.  Atmospheric science text books often include a cartoon to demonstrate how energy is absorbed and re-emitted as it passes through the atmosphere:



Here each arrow is 1 arbitrary unit of energy.  Things to note are that (1) each layer emits as much as it receives, (2) there's a lot more than 1 unit of flux within the atmosphere, (3) there's a steady 1 unit of net flux through the atmosphere and (4) this equilibrium is maintained by 1 unit in and 1 unit out at the top of atmosphere.