Which stations allow me to take my bike on board?
In central Paris, the gares de Magenta (RER E), de Lyon (RER A & D and regional services) and Saint-Lazare (regional services) are all cycle-friendly, fully equipped as they are with escalators and wide gates. Most stations beyond the périph' are also fine with cyclists taking their wheels on board. Note that the RER lines only allow non-fold-up bikes to be taken on trains on weekends.
What should I do in which season?
Spring: channel your inner Neville Longbottom and take botany classes at the Jardin des Plantes.
Summer: rent a paddleboat and join your fellow pedalophiles in Cergy-Pontoise or Boucles de Seine.
Autumn: go mushroom-picking, but with an expert – you don't wanna end up as one of the 1,500 chumps who get poisoned every year. More info here (in French).
Winter: try your hand at cross-country skiing in the forests surrounding Paris.
How can I share my walks with others?
Through the aptly named mapsharing community. Go on Google Maps and click on the 'My Places' button, followed by 'Create Map'. Once it's done and dusted, click the 'Share' button in the top-right. If you've found a nice route, publish it by all means – the walking community will thank you for it.