After my father died of cancer, I decided to begin researching cancer.
Literal
Father [subject-が] cancer [because of-で] passed-away [and-て], I [topic-は] cancer [about-について] research [object-を] begin-let's [quotative-と] thought.
ガンについての研究 uses について ('about') in its attributive form (についての) to modify 研究. 始めようと考えた combines the volitional (let's begin) with 考える (to think/consider) — a deliberate, considered decision.