Advanced brain imaging analysis methods, including multivariate pattern analysis (MVPA), functional connectivity, and functional alignment, have become powerful tools in cognitive neuroscience over the past decade. These tools are implemented in custom code and separate packages, often requiring different software and language proficiencies. Although usable by expert researchers, novice users face a steep learning curve. These difficulties stem from the use of new programming languages (e.g., Python), learning how to apply machine-learning methods to high-dimensional fMRI data, and minimal documentation and training materials. Furthermore, most standard fMRI analysis packages (e.g., AFNI, FSL, SPM) focus on preprocessing and univariate analyses, leaving a gap in how to integrate with advanced tools. To address these needs, we developed BrainIAK (brainiak.org), an open-source Python software package that seamlessly integrates several cutting-edge, computationally efficient techniques with other Python packages (e.g., Nilearn, Scikit-learn) for file handling, visualization, and machine learning. To disseminate these powerful tools, we developed user-friendly tutorials (in Jupyter format; https://brainiak.org/tutorials/) for learning BrainIAK and advanced fMRI analysis in Python more generally. These materials cover techniques including: MVPA (pattern classification and representational similarity analysis); parallelized searchlight analysis; background connectivity; full correlation matrix analysis; inter-subject correlation; inter-subject functional connectivity; shared response modeling; event segmentation using hidden Markov models; and real-time fMRI. For long-running jobs or large memory needs we provide detailed guidance on high-performance computing clusters. These notebooks were successfully tested at multiple sites, including as problem sets for courses at Yale and Princeton universities and at various workshops and hackathons. These materials are freely shared, with the hope that they become part of a pool of open-source software and educational materials for large-scale, reproducible fMRI analysis and accelerated discovery.