Classification/Regression Problem where Response Variable is Ordinal Data Science novice here!

I'm trying to work on the white/red wine quality data set, where I 'm trying to predict the quality of the wine. All the features are numerical. 

The response variable however, is ordinal with a quality score of integers 1 through 10 . I have seen tutorials try to group the scores ie: (0-4: Bad, 5-7: Good, 8-10: Great), but what if I wanted to predict the score as it is?

Should I use a regression approach where I try and minimize the error of my predicted scores versus actual scores? 

Or should I use a classification model anyways and instead of calculating a F-score to evaluate the model, find the model that minimizes a cost function?

Or perhaps there is another approach that works best?
You can treat this problem as multiclass classification and use classification algorithms to solve this problem. You can use any of the classification metrics like accuracy, precision, recall etc. 

Regression should be used only for the continuous data and continuous data is the one which can have infinite number of potential values within a given range. For example, for your problem the range is $1-10$, if it is continuous data, then it will have values like $1.04783,6.92838,8.2381,3.999,5.0$ etc. In this case you can opt for Regression algorithm.