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I suggest you start with the "Centro de Interpretação de Monsanto " (Monsanto Interpretation Center). It is a building with a lot of information about this area (pictures of flora and fauna, maps, guides...) and from there you can decide where you want to go next.
Unfortunately, this page is not translated, but you can try to use an automatic translater: http://www.cm-lisboa.pt/en/living-in/environment/monsanto-forest-park/espaco-monsanto From this page you can also grab more ideas.
Public transportations in Monsanto are not good. Bike is a very good option (generally speaking of course).