Model with two targets(classification, regression)? I have a dataset with features like Cylinders, Transmission, ExteriorColor, OdometerReading, StockType(New/Old), MakeModel, ModelYear, SoldYearMonth, CountSoldByMonth, Price, TopSelling.

Target column is 1 if a particular MakeModel is the top 5 selling vehicle in a month. So, given 60 months data I have to predict the next months top selling vehicles. I also have to predict the price go the vehicle. So I guess, I need to do both Classification and Regression .

MSRP   Cylinders   Transmission   ExteriorColor   OdometerReading   StockType     MakeModel            SoldYearMonth   CountSoldByMonth   Price   TopSelling
42098      8            MYC            Black           70348           Used    Chevrolet Tahoe            5/1/2019           10           39998     0
26390      4            MNK            Black            10             New      Buick Encore              5/1/2019           31           26390     1


and so on..
MSRP and Price are not always same.

How do I approach this problem?
You can also consider a multi-output model using the Keras functional API (see "Deep Learning with R", Ch. 7, p. 224). Here are the respective Keras docs. There are also some tutorials online. I'm not sure if a NN is suitable for your problem, but this would be an option to consider. The advantage is that your model can also take correlation between the features and the two outputs into consideration while learning (which may give better results).