That is a very wide question and the answer unfortunately is not simple, but very much dependent on your particular case, aim and priorities.
So, a two-class classifier. I would recommend creating a base learner using a logistic regression, then up the complexity with random forest and then finally you can try a neural network. For each step, you should record the performance and then you need to outweigh model performance versus model interpretability. Be careful with over-fitting, consider cross validation and think about how you record your performance and also which measure you use.
For consistent model comparisons, I would recommend looking into the mlr package, you can take a look at the official tutorial.
I hope this gets you started and good luck 