Hi Holli! For traveling from Bangkok to Bali, the easiest option is to fly. You can check Thai Airways, AirAsia or Garuda Indonesia for direct flights. All have multiple flights daily. Once in Bali, definitely spend a couple nights in Seminyak/Legian area - cute shops, bars and beach. Also check out Ubud for culture, temples and nature. 3 nights should be enough to see the highlights but it's easy to spend longer!
To get back to NY, I'd fly from Bali to mainland Asia like Singapore or Kuala Lumpur for easy connections to the US. Make sure to look a few months ahead for international flight deals. You could also consider extending your trip and stopping in another SE Asian country like Vietnam on the way back for extra adventures.
Solo traveling as a female in Asia is very safe as long as you use normal precautions. Hope you have a great trip!!