I was using htc Vivid and just switched to new One, and I was also confused about the Gmail settings as I can't remember if it was the same situation when I started using Vivid a while back.
Anyway, the solution is you need to switch on auto sync first, THEN go into the " Settings > Accounts & Sync" as everyone else suggested. Only after you start auto sync can you "untickle" each of the listed items (Calendar, Picasa, Google Play, etc.). From there you'll just need to leave Gmail tickled.
It is confusing because if you don't have auto sync on first, it won't show checkbox, as if you can only go into the menus again and again to do manual sync for specific app. But the idea I guess is that you have to grant auto sync authority first, then you can decide what not to sync, and uncheck them from the list.
Hope this helps.