You are right; #1 is correct. Talking is the gerund. "To talk" is the infinitive. You would say "I am talking," not "I to talk." I+verb+gerund.

What about this; I shy to talk in front of people. I think it's correct also.

In my opinion: "I feel shy talking in front of people" implies that you do talk in front of people, and the act of doing so makes you feel shy. "I feel [too] shy to talk in front of people" implies that you do not talk in front of people, and the mere thought of the possibility of doing so makes you feel shy.