I'd like to learn the basics of HirschSmale immersion theory. What sources are best for this? My background is mostly topological; however, many of the sources I've found on the internet focused on later work of Gromov on the hprinciple which seems more analytic than I would like.

M. Weiss has a very good survey on his homepage: http://wwwmath.unimuenster.de/u/mweis_02/papers.html called "Immersion theory for homotopy theorists". I also like very much M. Adachi's book "Embeddings and immersions". I think both are very good starters if you have a topological background. J. Francis has also some notes on his homepage: http://www.math.northwestern.edu/~jnkf/classes/hprin/ on a course about the hprinciple that can be helpful. Also Eliashberg and Mischachev's book "introduction to the hprinciple" is definitively a very good book. 


You may like the following lecture notes: Weiss, M. Immersion theory for homotopy theorists Francis, J. The hprinciple in topology 


Immersion theory has been "explained" by the Compression Theorem, with new proofs arguably being much more elementary and intuitive: 


I think this is the most readable source: Haefliger, A. Lectures on the theorem of Gromov. Proceedings of Liverpool Singularities Symposium, II (1969/1970), pp. 128–141. Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971. 

