network analysis: how to create nodes and edges files from csv
Decide how you want the graph to represent the data.  From what you've described one approach would be to have nodes in your graph represent People, and edges represent grants.  In that case, create a pairwise lis of people who are on the same grant.  Edges are bidirectional by default in iGraph, so you just need each pair once.
I have a two-mode (grant X person) network in csv format. I would like to create personXperson projection of this network and calculate some network measures (including centrality measures of closeness and betweenness, etc.). 
What would be my first step? I am guessing creating 2 separate files for Nodes and Edges and run the analysis in R using igraph package?!
Here is a super simplified version of my data (my_data.csv).

Grant,   Person  
A ,       1   
A ,       2
B ,       2
B ,       3