I would like to develop a small scale genomic portal that will (1) accept requests from the user; (2) search several tables of data (currently in the csv format); (3) perform computations in R; (4) output the result in the form of graphs.

This web app will be running on a remote web server (not my local machine) and it should be robust enough that could manage several user requests without breaking... The cBioPortal website http://www.cbioportal.org/public-portal/ is doing something very similar to what I need (but mine is in a much smaller scale)..

I know a lot of Python and R, but I have no idea where to start. I don't know any MySQL but I am willing to give it a go if I need.. Just not sure where to start. Any suggestions?
I would recommend to consider the following options, which will in the optimal case require very little programming:

 - Build your portal based on an existing CMS solution, first choice would be [Drupal][1] + [Tripal][2] depending on the the data in your files, assuming these are genomic data, they could possibly imported easily. The downside is that Drupal is in PHP in case you need to extend it directly. However, you could deploy small standalone python scripts for visualization. And most likely all desired functionality is already covered by extension modules.
 - Try to avoid executing heavy computation on the web-server, you need a compute backend of some sort with load balancing (a cluster or cloud solution)
 - If R is only for simple plotting I would try to avoid it, and check JavaScript visualization options, there are many good options, e.g. [google chart][3] or [highcharts][4].
 - If necessary you could evaluate a R/[Shiny][5] solution for visualization

Note: a beginner cannot implement a secure web-application (especially using DB backend) or set up a hardened server to run on, make sure to get help from a skilled sys-admin and programmer.

 [1]: https://www.drupal.org/
 [2]: http://tripal.info/
 [3]: https://developers.google.com/chart/
 [4]: http://www.highcharts.com/
 [5]: http://shiny.rstudio.com/