To model graph-structured data, graph learning, in particular deep graph learning with graph neural networks, has drawn much attention in both academic and industrial communities lately. The effectiveness of prevailing graph learning methods usually rely on abundant labeled data for model training. However, it is common that graphs are scarcely labeled since data annotation and labeling on graphs is always time and resource-consuming. Therefore, it is imperative to investigate graph learning with minimal human supervision for the low-resource settings where limited or even no labeled data is available. In this tutorial, we will focus on the state-of-the-art techniques of Graph Minimally-Supervised Learning, in particular a series of weakly-supervised learning, few-shot learning, and self-supervised learning methods on graph-structured data as well as their real-world applications. The objectives of this tutorial are to: (1) formally categorize the problems in graph minimally-supervised learning and discuss the challenges under different learning scenarios; (2) comprehensively review the existing and recent advances of graph minimally-supervised learning; and (3) elucidate open questions and future research directions. This tutorial introduces major topics within minimally-supervised learning and offers a guide to a new frontier of graph learning.