Graph theory is one of the most important topics in discrete math and programming. According to me, the most crucial step in solving graph theory problems is visualising them properly.

Most people will tell you "Go to XYZ website, sort problems by graph theory and start practicing". This has to be done any way for any topic. Let me tell you something more interesting. In this answer, I shall try to throw light on how you build intuition about graph problems and how you can know which approach to take. Let's go step wise:

Read the question very carefully for determining what type of graph it is:

This is the first and probably one of the most important steps. The question will tell you whether the graph is a tree, or it is a graph with only one cycle, or multiple disjoint cycles, or just a random graph. A tree has so many nice properties that you can exploit. There are subtle ways of saying that the given graph is a tree. For example: "The graph has N nodes. It is not disjoint and has N-1 edges", "The graph doesn't contain any cycles", "In the given directed graph, each node has one and only one parent", etc. Similarly, if there is only one cycle, then you can imagine the graph as having a ring with many trees hanging from the nodes of the ring.

Once the graph has been identified, identify what the question wants from you:

In this step, you try to figure out what is it that the question wants you to do. Does it have something to do with shortest paths? If yes, then are all the weights positive? If not, then clearly it can't be direct Dijkstra. What else can we do? Bellman Ford? Floyd-Warshall? You should be able to eliminate approaches fast.

Graph problems can also include simple traversal, flows, disjoint unions, etc. The best way to figure out what is applicable is method of elimination. The size of input data tells a lot about the algorithm to be used. This is because graph algorithms span the whole range of complexities, from linear to exponential. So, if you know you need a linear or a quasi-linear algorithm for the given size of input, you can simply eliminate many of the algorithms.

If you have identified that the graph is not a tree, a lot of algorithms can further be eliminated like heavy-light decomposition, dynamic programming on trees, centroid decomposition, etc, unless the question allows you to form a tree of biconnected components.

Coming up with an algorithm to solve the problem:

Once you have identified what the question is asking you to do, and what is the basic algorithm that is going to be used, the last step is to come up with a working solution. The best way according to me is to solve the problem with pen and paper. You construct small graphs of the kind that the question puts out, and you try to apply the techniques that you shortlisted from the previous section. This is because graph algorithms generally are quite good at scalability. What this means is that if you observe a particular property for a small graph that you draw on paper, it is highly likely that the property will be true for similar graphs of any size . So, by trying out techniques on paper, you can get to a solution in most of the cases.

Coding graph problems:

Graph problems are generally code intensive. But they follow a very standard pattern. This is the point where practice comes in. You need to code quite a few graph problems before you reach a stage of mastery. The biggest hurdle is coding what you have visualised since graph problems tend to be more abstract in nature than something like data structures or greedy algorithms. There is only one way to work around this hurdle: practice. The more you code, the better you become with putting your visualisation on to your editor.

This is all I can think of at this point in time. I shall edit this answer if I feel I missed something.

All the best :)