There isn't any specific verb for work out.You can say : iść na siłownię, poćwiczyć na siłowni ( albo w klubie fitness ), (po)trenować.
So: Chodżmy poćwiczyć na siłowni.
Chodżmy do klubu fitness.( it is obvious for what )
Może dzisiaj potrenujemy? ( suggestion )
From the above options I like Idziemy na siłownię?
Seems the most natural to me.
As for the fitness klub, I must admit it isn't often used in my neck of the woods, and it sounds somewhat pretentious to me, a kind of word you're more likely to hear in a commercial or see in a brochure advertising a place of this type. I used to work out at a gym which included a fitness club too, but no one reffered to it as fitness klub, even the people who used this facility. I sometimes heard them saying idę na fitness, but I wouldn't since it doesn't make any sense.
As for the occurence you could use both depending on the context.