Wandlebury Country Park - click the name for more info - which is not far from Cambridge and can be reached by car, bike or bus. It used to be a stately home and the grounds are breathtakingly beautiful. This photo was taken by the pond, which is full of fish, frogs, shimmering dragon flies... You can get a map with trails for 50p and if you arrive by car you pay £2 for the parking (come by bus or by bike and you just pay for the map). Be sure to pack a picnic as we didn't find any food outlet there. This is such a small outlay for the opportunity to spend a wonderful day outdoors. If you feel really adventurous, the longest trail meets the Roman Road, so if you have a map of that you can walk on for miles... A new cycle track is opening that allows you to cycle from Cambridge.
Cambridge Past, Present and Future also manages other great attractions, I recommend the Coton Countryside Reserve, which is totally free (including parking).
Milton Country Park - click name for more info - which offers lovely walks, a play ground for all ages and a cafe. If you come by car you pay £2 for all day parking, if you come by bike or by bus it's free. The area is not as big as Wandlebury but if you have a small baby you might like having more facilities at hand in a more compact space.
Do pay a visit to these wonderful places and let me know how you liked them...