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I'm very similar to you. except for the fact that I grew up in Australia and not Canada. My parents only spoke English to me and I felt like I was losing a huge part of my culture by not knowing Tamil.
A few things I recommend:
Learn Tamil. Check out Tamil Language in Context and practise speaking with your family, or if you're too nervous you can take conversation lessons on Italki. I also started a YouTube channel called Learn Tamil Through English where I try to create videos that are helpful for diaspora Tamils/people married to Tamils/anyone interested in the history and literature.
Specifically for Sri Lankan Tamils, I'd recommend Tamils in Sri Lanka: A Comprehensive History (C. 300 BC - C. 2000 AD), a book by Murugar Gunasingam. The first few chapters are a bit slow, but it picks up in the Cholas chapter and it will teach you a lot about your history.
Good luck mate. Let me know if you need any help.