Well, they have only studied mRNA vaccines for other purposes (mainly cancer treatments) for thirty years. So yes it is a very new treatment but it has been studied for a long time, with clinical trials of other mRNA vaccines (mostly for melanoma) going back several years.
I agree that there is a possibility of long-term effects, but that possibility isn't very large. So far these vaccines have not just proven to be safe, but to be extraordinarily safe. Your odds of dying of covid are approximately one thousand times higher than dying from this vaccine.