You can literally look at a map of Canberra and pick just about any green patch for a picnic (after checking you can get inside), but if you want more location-inspired topics of conversation than, "Hey look! Grass and stuff!" I recommend the Australian National Botanic Gardens. It's easy to drive or bus there (although the parking is not free) and if you want a really special time you can check events on their website.
The downside: It needs to be a nice day, and comfortable shoes are a must if you want to go further than the cafe.
The National Gallery always has a wide range of exhibits, but my favourite part is the Sculpture Garden, especially when the fog sculpture is on. From the Sculpture Garden, you can wander down to the shores of Lake Burley Griffin and enjoy an exquisite wander.
You can have an equally exquisite wander from the other side of the lake, starting from The National Carillon. Once again, your day will be greatly improved by good timing, since the Carillon plays often (but not constantly). Both of these locations are great for picnics (or just ice creams), too.
The downside: Not everyone likes art or music or walking.
1. Picking Your Moment
Canberra bursts with high-quality free events, particularly this centenary year. In my opinion the two best are Skyfire in Autumn (16 March) and Floriade in Spring (14 September-13 October). Floriade can be overwhelmingly crowded, so I recommend a picnic near the Commonwealth Park Playground before plunging in, especially if you'd like a subtle way to gauge your date's feelings about children.
The downside: These events are extremely popular, and the parking is painful, especially when the event is finished. On the other hand, tense moments in the car are a great way to get to know your date.