"Since then" is your best bet in this instance. It refers to a period of time starting somewhere in the past and reaching up to this very instant.
"I met my wife 34 years ago. Since then, I've been in love with her." This is good, common English.
"Onward" and "onwards" are very similar to each other and, in my experience, can essentially be used interchangeably. However, in American English, this isn't a common expression. Unlike "since then", "onward" refers to a period of time starting now and extending into the future. So, it is quite different.
"From this day on" is like "onward" in that it refers to moving into the future.
"From that time" seems awkward to me. I'd recommend "since then" or "since that time" instead.
"Once, I touched a hot toaster. Since then, I certainly haven't done it again. I learned my lesson well."