I usually have a rough idea in mind of what I want to get people and wait until the sales to start shopping. If what I was going to buy is on sale, great! If not I buy it anyway.
I've given up on Christmas gifts. My family isn't Christian, so we just give $$$ in red envelops for Chinese New year =) So Christmas gifts are usually just for my friends, and since the exchange of gifts is usually centered around a family event, usually I'm not involved in these. I focus on my friends' birthdays instead.
A few times some friend groups have tried to organize a secret santa sort of event, but now that we're all losing touch a little bit since we've moved all across the country, it's harder to ensure that everyone keeps in touch with everyone in the group.
It tends to be towards the end of November, start of December when I buy Christmas gifts for other people. I generally just wait until other people tell me what they want, then I'll go and buy it, preferably while Black Friday sales are on, lol.
I always try to start early, but it never seems to work out in my favor like that. I usually end up doing it later because this time of the year gets super busy for us all and there's always something else that needs immediate attention.