If your main priority is an active and outdoor-focused lifestyle then you should consider moving to the suburbs of Frankfurt.
Most of those small towns aren't that far away as Frankfurt is not a megacity and it will take you about 30 minutes by car or public transportation (the infrastructure to the suburbs is pretty good) to get to the city center.
But remember, Frankfurt is not only an expensive city but also an expensive are meaning that the suburbs can be quite expensive, too and I am not talking about accomodation only but the whole living expenses.
For an outdoor person the Taunus area would be best but it is also the most expensive area and finding a flat there can be pretty tough. If you move further away in direction of Hanau there is a lot of countryside but infrastructure isn't that good anymore so you will definitely need a car to travel to Frankfurt. And countryside can be boring by the way.
You should make a tradeoff between price, distance to the city, "outdoor" sights and range of activities.
In Frankfurt there aren't that many places to walk with your dog and it can be expensive but on the other hand you have a wide range of other activities. On the countryside you will need a car and it can get boring if you don't drive to Frankfurt but it is much more affordable and there is space for outdoor activites. But the closer you are to Frankfurt the more expensive it gets, especially in the Taunus area which is a nice and beautiful area.