8. Ne5 doesn't really accomplish anything. And as pointed out 9. g3 just weakens your kingside. When your opponent brings their queen out early, you can get an advantage by making moves that attack the queen and simultaneously do something good for your position. But moves that just attack the queen but do nothing for your position or weaken it, don't help.
I prefer 8. d4 or 8. Re1. Continuing to develop and maintaining the possibility of a discovered attack by moving the knight later. Notice that 8. Ne5 Qh4 9. Bh5+ g6 10. Nxg6 hxg6 11. Bxg6+ Kd8 12. Re1 Be7 doesn't quite work because you don't have enough development. So continue to develop your pieces and then look to unleash an attack.
12. Bd3 This attacks the queen, but doesn't gain much for you. An interesting idea here is 12. Bc4. Then you are threatening Qe2+ with a double attack on the a6 knight. If 12. ... Nb4 then you have 13. Nh4 and black can choose between
i) 13. ... Qxc2 14. Qh5+ g6 15. Nxg6
ii) 13. ... Qa4 14. Nd5 threatening both Nxc7 and Qh5+
I haven't examined all these ideas carefully, but there are some ideas to chew on.
It seems I started the game pretty well, with what looks like a bit of advantage by the end of move 14. I played White and tried to chase the queen around and get the development done at the same time.
I ended up loosing the game. Where do you think I went wrong? What was the biggest mistake?