Sending money to Russia for family back home

I have heard that I can give 10k as a gift in cash for my aunt to take on the plane.

Please don't, for her own safety. Don't know when was the last you've been to Russia, but that's not a place to walk around with $10K in cash in your pocket.

For the rest, the 20k, I am not sure what is the best course of action. Would something like Western Union, Paypal or Bank Wire Transfer be the best course of action?

Wire transfer would be the safest option.

Would there be tax implications for me as well?

Depends on where you are tax resident and where you are a citizen. Some countries have "gift tax", but most don't.

If you're a US tax resident, then you're subject to US gift tax rules. Your gifts are taxable if they exceed $14K per year per person. So your $30K to your mom is taxable. But your $10K to your mom, $10K to your dad and $10K to your aunt is not. You cannot however control what they do with it.