How do I charge CGST and SGST in my state and IGST elsewhere?

Yes, but it helps to know one rule: at checkout, every rate that matches the customer's address is added together (the hint on the page talks about the most specific match winning, but rates actually stack).

In Settings → Store Settings → Manage Tax:

1. For your own state, add two rates with Country India and State Tamil Nadu: one with Name CGST and Rate 9, and one named SGST at 9. Customers there see both lines.
2. For IGST, add a rate of 18 named IGST for each other state. Don't add an India-wide 18% rate, because it would also match Tamil Nadu and stack on top of CGST + SGST.

The same setup works for US state sales tax or regional surcharges. Adding many rates is quicker with Import, which needs the Import/Export plugin. Your plan may limit how many tax rates you can add.