I have a 2yo boy and we live in Fremont. I take him on lots of walks and to different parks and playgrounds. Lake Elizabeth has the stream that someone already mentioned. It also has the large lake, several playgrounds, ducks, geese, lots to see and do. There is a small pond between the soccer fields and the big lake, and that is fun to look at too (something different).
There are few nice parks in the Niles area of Fremont. If you park at 3rd street park, there is a playground, and access to the creek and ponds. There are ducks, and lots of turtles that live in the ponds which you can see from the paths. It's really fun for the kids to see.
On Walnut is a place called Tule Ponds at Tysen Lagoon. I'm not sure the hours, but they have shceduled activies...swing by and check it out, or go to msnucleus.org. Byington Park in Newark is fun on a hot day (there are water features for the kids to play in).
The libraries are great. Check out the Alameda Co. website for dates and times of library activities...
A fun, close, easy day trip is to Happy Hollow park and zoo in San Jose. It's really only a 20 minute drive if you miss traffic.
Other fun trips - lots of beaches if you want to go to the coast. If not the coast, my son likes the Berkeley Marina. They can see boats, bridges, birds and other wildlife, people flying kites, all kinds of stuff. I haven't taken my son, but I used to go to Coyote Hills as a kid. During the fall, be sure to check out Ardenwood Farm for pumpkin patch, hayrides, stuff like that. Also, Take her for a ride on the train that runs between Niles and Sunol. This is especially popular during the holidays, but would be fun all year round. Oh, I've also heard Shinn Park is a great place in Fremont. They have a rose garden, which the kids usually enjoy. If you want to travel a little more, try the Oakland Zoo or Tilden Park.
Lots to do...I've lived here my whole life...Hope these suggestions help!