I am not so experienced but I think you can approach learning about graphs and data-structures in this way:

You can start learning data-structures but while you are at it you can improve your implementation skills while practicing DIV 2 Problem A and Problem B and sometimes C too.

Important data structures which you should start with are the (Basic), you could just implement them using STL (in C++) or other library (of your language)[Except Tree]. You can also explore other elements of STL like map and set now from Topcoder tutorial : https://goo.gl/x42wxm

Data structures (Basic): Stacks, Queues, Trees and Heaps. Data structures (Advanced): Trie, Segment trees, Fenwick tree, Disjoint data structures

Then when you have theoretical knowledge of graphs and other data structures then you can start implementing them and when you are able to implement it properly you can search those questions which have tags of those data-structures and solve them. For Graph algorithms, you need to solve a lot of simple DFS and BFS based application questions and then move on to other algorithms like LCA, Union Find etc.

Here is a good collection of links for learning Data-structures : https://goo.gl/eI59Rn Hope you enjoy learning Data-structures. ;)