Hmmm...it's hard to do entire fields justice, but here's an attempt.
There are a couple of standard neurobiology textbooks, like Kandel, Jessel, and Schwartz's Principles of Neural Science, Purve et al.'s Neuroscience and Squire et al.'s Fundamental Neuroscience. These are huge books that cover a bit of everything, and you should know that they exist, but I wouldn't necessarily start there.
If you're specifically interested in computation, I would start with David Marr's Vision. It's quite old, but worth reading for the general approach he takes to problem-solving. He proposes attacking a problem along three lines: at the computational level ("what operations are performed?"), the algorithmic level ("how do we do those operations?"), and the implementation level ("how is the algorithm implemented").
From there, it depends on what you're interested in. At the single-cell level, Cristof Koch has a book called The Biophysics of Computation that "explains the repetoire of computational functions available to single neurons, showing how individual nerve cells can multiply, integrate, and delay synaptic input" (among other things). Michael London and Michael Häusser have a 2005 Annual Reviews in Neuroscience article about dendritic computation that hits on some similar themes (here: https://www.researchgate.net/publication/7712549_Dendritic_c... ), along with this short review (http://www.nature.com/neuro/journal/v3/n11s/full/nn1100_1171...) by Koch and Sergev, and a 2014 review by Brunel, Hakim, and Richardson (http://www.sciencedirect.com/science/article/pii/S0959438814...). Larry Abbott has also done interesting work in this space, as have Haim Sompolinsky and many others. Gordon Shepard and his colleagues maintain NEURON (a simulation package/platform) and a database of associated models (ModelDB) here: https://senselab.med.yale.edu/ if you want something to download and play with (they also do good original work themselves!)
Moving up a bit, the keywords for "weight adjustment" are something like synaptic plasticity, long-term potentiation/depression (LTP/LTD), and perhaps spike-timing dependent plasticity. The scholarpedia article on spike-timing dependant plasticity is pretty good (http://www.scholarpedia.org/article/Spike-timing_dependent_p... Scholarpedia is actually a pretty good resource for most of these topics. The intro books above will have pretty good treatments of this, though maybe not explicitly computational ones.
More to come, however I also just found this class from a bunch of heavy-hitters at NYU: http://www.cns.nyu.edu/~rinzel/CMNSF07/ Those papers are a good place to start!