It depends on the school you apply to, but it's usually not a big deal to have one failure in first year. I am making the assumption that by "first year" you mean literally first year - as in this is your first year in university and last year you were in high school.
I can speak with confidence only from a Canadian perspective, so please see what others say from around the world. The vast majority of schools in Canada require you to maintain a certain average in your final two years of study in your undergraduate program, and the first two years are not considered. I have at least two colleagues who failed more than one course in their undergrad first and second year and were still admitted. See here as an example of what these admissions requirements are.
Doing a cursory glance at some international schools, the "last two years" thing doesn't appear to be universal but the admissions requirements are. For example, TU Delft in the Netherlands requires a minimum 75% overall GPA and "scores in key subjects to be good", which I presume is dependent on the program see here. No mention on failing grades and no inadmissibility is mentioned due to a failure.
Since it depends so much on where you apply, I can't say with 100% confidence that you will not have any troubles with your F. However, I can certainly assure you that you are not the first person who failed something in first year and still got in to grad school.
A more serious question you need to solve for yourself, probably with help from your academic advisors if they exist where you are, is how to recognize and prevent this from happening again. Failing a course basically means that there is a pattern of behaviour throughout a school term that led you to that state. Unless your writing course had a single, 100% final, it's unlikely that you were surprised by the failure. Recognizing the patterns early will help to prevent this again.