You're asking for a short answer to an entire field of research. You won't get a comprehensive one that fits in a SE answer.
Generally, yes the genes/their products (which are often treated as the same thing in these abstracted models) are the nodes; the causal relationships between them are coded in (typically weighted, directed, bidirectional) edges. In graph theory more generally, you hold the "activity" of the nodes in a separate vector; sometimes that is taken as an input vector to your graph to produce a corresponding output vector at some time dT in the future (other times you aren't really interested in the activity at all and are working with just the graph itself).
Edge weights are determined by experiment, often many many interrelated experiments. If you experimentally increase or decrease expression of gene X, you can monitor gene Y for change to infer a "connection". However, it's more complex than that because an effect of X on Y could be mediated through a number of other nodes. Biology is rarely simple, and if you expect it to be you're going to make mistakes rather quickly.
If you have enough information about the graph and timeseries data about expression, you may be able to infer a model graph to describe the observed data. Some people have already compiled putative networks (particularly in model organisms) and you can find them by searching for their published papers describing them, or finding others who cite them.
If you want to do these things practically, I highly discourage working merely from textbook examples or broad summary reviews; to do something useful in science, you need to read primary literature. Your answers to these questions in your particular area of study will come by reading the methods sections of papers that publish on these networks.
Some references to start with:
Emmert-Streib, F., Glazko, G., & De Matos Simoes, R. (2012). Statistical inference and reverse engineering of gene regulatory networks from observational expression data. Frontiers in genetics, 3, 8.
Lee, T. I., Rinaldi, N. J., Robert, F., Odom, D. T., Bar-Joseph, Z., Gerber, G. K., ... & Young, R. A. (2002). Transcriptional regulatory networks in Saccharomyces cerevisiae. science, 298(5594), 799-804.
Pavlopoulos, G. A., Secrier, M., Moschopoulos, C. N., Soldatos, T. G., Kossida, S., Aerts, J., ... & Bagos, P. G. (2011). Using graph theory to analyze biological networks. BioData mining, 4(1), 1-27.
Vijesh, N., Chakrabarti, S. K., & Sreekumar, J. (2013). Modeling of gene regulatory networks: A review. Journal of Biomedical Science and Engineering, 6(02), 223.
