I am wondering if I should correct the Sentinel 2A L1C data for atmosphere and covert it to L2A data or use it as it is. Also, what should be done to convert the data?
The answer to your question depends mostly on two premises: (1) what exactly do you want to achieve? what is your goal?, and (2) what is your expected level of accuracy? to what extent does your objective depend on the accuracy of your input data? In the absence of information about these points, it's difficult to provide a very specific answer to your question. Nevertheless, here is a generic recommendation:
Atmospheric correction is essential if you need to generate accurate information or reliable services, for instance. If that's the case, and/or if you are not familiar with the atmospheric correction process, you should directly acquire Level 2 (i.e., atmospherically corrected) products, in which case you will not need to worry about that step. Look at these web pages for further information:
On the other hand, if you are specifically interested in learning about or contributing to the complex process of separating the atmospheric from the surface contributions to the radiance measurements acquired in space, then you first need to study the theory of radiation transfer in absorbing and scattering atmospheres. By the time you will have built experience and expertise in that area, you will also have gained the understanding that atmospheric corrections are always required for quality work, and the skills to implement those corrections in way appropriate for your purpose.
Michel's explanation is spot on, particularly regarding the correlation between the specific goals of your project, the required accuracy and the need for at. correction. I frequently use the Sen2Cor plugin for SNAP (Sentinel Application Platform) software developed by ESA (European Space Agency). This plugin performs an atmospheric correction from TOA to BOA by adapting ATCOR.