Rent bikes and bike around Stanley Park. Can pack a picnic lunch. It's wonderful.
Depending on how fit you are, you can hike up Grouse Mountain and do the same (instead of paying for the gondola - you used to be able to ride it down for free).
I second the grouse mountain -- even if you pay the passenger rate to get to the top, the value on a semi-clear day is definitely there. Pack a lunch or have a drink at the bar with the great view. Take Transit to get there to avoid paying gas, parking. You could make a day trip of it, and stop off mid-way on the skytrain for some ethnic foods (Richmond Chinese food, for example).
The skytrain / sea bus is a nice view and a way to go for a tourist. Check out granville island or walk around downtown to look at the art galleries, there are several excellent ones.
Fish and Chips in White Rock.
Rent kayaks in Ladner for a half day.
There may be some winery / farm tour activities still open, but I don't know which ones are seasonal.
There are a couple of heritage farmhouse places to visit, not too far, if you are into history (Surrey / Delta). Or even the Burnaby Village Museum.
My teenage son would strongly recommend the indoor go karts in Richmond. Not "free", but I don't know your budget.