'Exercise', meaning physical exertion, is an uncountable noun, and therefore singular. It's often preceded by 'some'. For example, 'I'm not fit enough - I need to do some exercise' or 'I sit all day behind my desk, so I like to get some exercise in the evenings.' 'Exercises', as a countable/plural noun, are repeated tasks, with the goal of practising, learning or improving something. You can have grammar or maths exercises at school, but you can also have physical exercises. For example 'The coach gave the team members exercises to do to improve their strength'. This might be repeated bends, stretches and so on.

Exercise and workout are equivalent. You can say I'm going to go exercise or I'm going to go workout.

Also, if you just going to the gym in general you would use exercise. The only time you would use exercises is if you were talking about doing more than in succession. Hope this helps