They actually carry very different suggestions to me. "We have started nothing!" tells me you planned to start something, but it was a complete failure. You are exasperated. "We started nothing!" tells me this is a denial of responsibility. You are not to blame, although someone is blaming you. This is not a really good example to compare simple past and present perfect. I'd suggest finding some online exercises so you get used to the difference.

In that case, you would use the present perfect: "We have started nothing." I means from the past until right now. Use the simple past ("We started nothing") to talk about something or some time period that is completely finished (e.g., "We should have started last year, but we started nothing, and now it is too late").