Well, if you're looking for a textbook, I suggest Biopsychology by Pinel. It was required reading for two courses I took, and I found it to explain things quite well.
Something to watch out for though: he tends to over-simplify things in places, which may lead you to develop overly-rigid mental models. There are exceptions to every rule, and some recent papers may contradict what Pinel says about the functions of some areas of the brain, or show that things are actually more nuanced. It's still a good start.
Once you have a good grasp of the fundamentals, I suggest reading Nature Reviews Neuroscience. They often have excellent review articles that summarize the state of research in a particular area. http://www.nature.com/nrn/index.html
If I were to suggest one area to focus your attention on, it would be memory. Memory is a central ability of the mammalian brain, and pretty much every area of the brain is involved in memory somehow. Here are some papers that I've found enlightening:
Gaffan. Against memory systems. Philosophical Transactions of the Royal Society B: … (2002)
Martin and Morris. New life in an old idea: The synaptic plasticity and memory hypothesis revisited. Hippocampus (2002)
Frankland and Bontempi. The organization of recent and remote memories. NATURE REVIEWS NEUROSCIENCE (2005) vol. 6 (2) pp. 119-130
Ross and Eichenbaum. Dynamics of hippocampal and cortical activation during consolidation of a nonspatial memory. Journal of Neuroscience (2006) vol. 26 (18) pp. 4852
Dudai. The Neurobiology of Consolidations, Or, How Stable is the Engram?. Annual Review of Psychology (2004) vol. 55 (1) pp. 51-86
Fischer et al. Recovery of learning and memory is associated with chromatin remodelling. Nature (2007) vol. 447 (7141) pp. 178 (but don't worry too much about the parts that are heavy on the biochemistry).
Rasch et al. Odor cues during slow-wave sleep prompt declarative memory consolidation. Science (2007) vol. 315 (5817) pp. 1426
One thing you may notice is that all of these papers are published in the last decade. That's about the shelf-life of research in this field. Let me know if you can't find these articles online; I have PDF copies of all of them.
Once you have a good grasp of the fundamentals, I suggest reading Nature Reviews Neuroscience. They often have excellent review articles that summarize the state of research in a particular area. http://www.nature.com/nrn/index.html
If I were to suggest one area to focus your attention on, it would be memory. Memory is a central ability of the mammalian brain, and pretty much every area of the brain is involved in memory somehow. Here are some papers that I've found enlightening:
Gaffan. Against memory systems. Philosophical Transactions of the Royal Society B: … (2002)
Martin and Morris. New life in an old idea: The synaptic plasticity and memory hypothesis revisited. Hippocampus (2002)
Frankland and Bontempi. The organization of recent and remote memories. NATURE REVIEWS NEUROSCIENCE (2005) vol. 6 (2) pp. 119-130
Ross and Eichenbaum. Dynamics of hippocampal and cortical activation during consolidation of a nonspatial memory. Journal of Neuroscience (2006) vol. 26 (18) pp. 4852
Dudai. The Neurobiology of Consolidations, Or, How Stable is the Engram?. Annual Review of Psychology (2004) vol. 55 (1) pp. 51-86
Fischer et al. Recovery of learning and memory is associated with chromatin remodelling. Nature (2007) vol. 447 (7141) pp. 178 (but don't worry too much about the parts that are heavy on the biochemistry).
Rasch et al. Odor cues during slow-wave sleep prompt declarative memory consolidation. Science (2007) vol. 315 (5817) pp. 1426
One thing you may notice is that all of these papers are published in the last decade. That's about the shelf-life of research in this field. Let me know if you can't find these articles online; I have PDF copies of all of them.
Hope that gives you some direction :)