No, neurogenesis is not necessary for learning (assuming you are talking about adult neurogenesis: of course you need to generate neurons sometime in development to even have a nervous system).
Easily the most-accepted theory for learning and memory is that learning happens by changes in the strength of synapses, the connections between cells. The strengths of these connections are also often referred to as "synaptic weights." The primary mechanisms for this learning are long-term potentiation and the related spike-time dependent plasticity, as well as the guiding principle of Hebbian learning often stated as "cells that fire together, wire together."
Briefly, long-term potentiation or LTP describes the increase in post-synaptic receptors that occurs under certain conditions: this is what increases the efficacy or strength of certain synapses. One of the key conditions is that LTP tends to occur at synapses that causally contribute to making the post-synaptic cell fire.
Adult neurogenesis primarily occurs in a few localized area of the brain (Zhao et al.,
2008). Although it is possible there is neurogenesis outside of these areas, it has not been demonstrated conclusively and is certainly not sufficient to be necessary for learning in a normal, uninjured adult brain.
Neurons born in the subventricular zone continually replace olfactory receptors, which have a relatively short lifespan, probably evolved due to their constant exposure to all sorts of potentially damaging external compounds entering the nose. These cells do have to undergo some "learning" processes to integrate into existing networks, but that isn't what you would colloquially think of as learning in general.
The best candidates for cells that actually impact learning to some degree are the new neurons born in the subgranular zone of the dentate gyrus in the hippocampus. The hippocampus is indeed an important structure for certain types of learning, but new neurons aren't necessary for this learning. Although it is still somewhat unclear what the roles are of the new neurons in the dentate gyrus, it is thought that they contribute to the primary roles of the dentate gyrus, which are to help distinguish between similar patterns of activity (Deng et al., 2010), also called "pattern separation."
Two other hypotheses are that adult neurogenesis helps to encode timing information in memories because the new neurons only participate in new memories, or that they allow for incorporation of novel stimuli once the older neurons have been conditioned and tuned only to stimuli that have been experienced before (Aimone, et al., 2013). There is evidence for all of these hypotheses but none of it is conclusive, and all may be true to some degree.
There are many different types of learning and these different types of learning can occur in different brain regions, most of which do not demonstrate any substantial adult neurogenesis. Often people think about associative learning with operant or classical conditioning. There is also episodic learning: remembering events that have happened (this is the type most closely associated with the hippocampus, where perhaps neurogenesis plays some role). There is also motor learning, and several types of learning in sensory systems like habituation or sensitization. Therefore, it is important to understand that learning isn't just one thing, and although the mechanisms can be related they are not always the same.
How to learn more: I can tell you about courses at institutions I have been affiliated with, but of course these could vary between schools. Learning and memory is usually at least part of a systems-level undergraduate neuroscience course, which is often the second of two overview courses in neuroscience. The most common time for students to take such a course would be as a second-semester junior or a second-semester first-year graduate student, because there is prerequisite knowledge in physics, chemistry, molecular biology, and physiology. There are also graduate-level courses in the molecular mechanisms of learning and memory, and theoretical neuroscience courses that cover the computational considerations of learning and memory rather than just the biological mechanisms. These types of courses might be available to a senior-level undergraduate as well.
Of course you can also read some of the review papers I have included here and follow the references they cite for original research, as well. The Aimone review is particularly readable and has some very excellent illustrations for a novice, and also extensively covers the regulation of neurogenesis by behavior and environment.
References
Aimone, J. B., Li, Y., Lee, S. W., Clemenson, G. D., Deng, W., & Gage, F. H. (2014). Regulation and function of adult neurogenesis: from genes to cognition. Physiological reviews, 94(4), 991-1026.
Deng, W., Aimone, J. B., & Gage, F. H. (2010). New neurons and new memories: how does adult hippocampal neurogenesis affect learning and memory?. Nature Reviews Neuroscience, 11(5), 339-350.
Zhao, C., Deng, W., & Gage, F. H. (2008). Mechanisms and functional implications of adult neurogenesis. Cell, 132(4), 645-660.
