I am seeking a basic list of key data analysis methods used for studying social media platforms online. Are there such key methods, or does this process generally vary according to topic? And is there a standard order in which these methods are applied?(The particular context I'm interested in is how the news is impacting on social media)
The kinds of tools you will use will vary based on the problem you are trying to solve. Social media data is rich and therefore many questions can be asked - and many tools can be used.
However, there is a general pattern you might keep in mind. Typically, you will have to use the platform's API to gather data. You will then have to normalized and store the data in a data warehouse. Finally, you will access and analyze the data with the tools you desire. Keeping the end goal in mind, you will have to strategically choose the best technologies for the job you are doing.
For example, let's say you wanted to study the relationships between users on a social network - a question like "Who are the mutual friends of the most popular individuals in this social network?" In this case, you would gather data using the social media platform's API, normalize it into a CSV, import it into a Neo4j database, and then use Cypher to make queries.
You might want to try this book Mining the Social Web for an overview of different techniques. Obviously, the methods you need will depend on the use case. A lot of people do interesting things with graphs, displaying relationships between users, with respect to certain topics. Or you might simply to a timeline showing how a news topic builds in interest and wanes.