Chad Topaz's page on Getting Started with Topological Data Analysis is a great place to start. It includes a list of papers that are good for newcomers to the field as well as exercises to learn some of the topology basics.
Textbook: Computational Topology for Data Analysis by Tamal Dey and Yusu Wang (PDF available for free)
Survey Paper: Topological Data Analysis via Undergraduate Linear Algebra by Cheyne Glass and Elizabeth Vidaurre
Survey Paper: A roadmap for the computation of persistent homology by Nina Otter, Mason A Porter, Ulrike Tillmann, Peter Grindrod and Heather A Harrington
Short Introduction: What is ... Persistent Homology? by Shmuel Weinberger
Atmospheric Science: Using Persistent Homology to Quantify a Diurnal Cycle in Hurricane Felix by ST, Elizabeth Munch, Jason Dunion, Kristen Corbosiero, and Ryan Torn
Resource Coverage: Persistent Homology for Resource Coverage: A Case Study of Access to Polling Sites by Abigail Hickok, Benjamin Jarman, Michael Johnson, Jiajie Luo, and Mason A. Porter
Audio Analysis: Topological Fingerprints for Audio Identification by Wojciech Reise, Ximena Fernández, Maria Dominguez, Heather A. Harrington, and Mariano Beguerisse-Díaz
Spatial Applications: Spatial applications of topological data analysis: Cities, snowflakes, random structures, and spiders spinning under the influence by Michelle Feng and Mason A. Porter
Biological Patterns: Topological data analysis of zebrafish patterns by Melissa R. McGuirl, Alexandria Volkening, and Björn Sandstede
Applied Algebraic Topology Research Network (AATRN). You can subscribe to get emails about upcoming seminars. The AATRN youtube channel has hundreds of recorded talks on topics in various areas of applied topology.
Women in Computational Topology (WinCompTop). Hosts events and has a mailing list where lots of opportunities are advertised.
DONUT. A repository of papers from applied topology.
Note: this is a very incomplete list! If you have a resource you think should be listed here let me know!