A good activity to do is a walk along the Yarra River. There are a few bridges across the river, so you can make the walk as long or as short as you want. I will describe here the walk that we did along the river, bringing us to a few gardens and sights, and offering some spectacular views on the city's CBD.
We started at the nice Flinders Street Station, walking along Flinders Street eastwards to Queen Street for 300m. Going south, you will cross the Queen's Bridge across the Yarra. Just on the south bank below the bridge is the Tan Tan trail, a beautiful biking and walking trail all along the Yarra River. We took the trail westwards, where you will pass the Southgate footbridge. 300m further, yet another bridge leads back into the CBD. Beyond this bridge starts "The Domain", a huge and beautiful park area (around 100ha if you include Royal Botanical Gardens). It's a very nice area to walk and there are many trails leading in different directions.
The northern part of The Domain consists of the Alexandra Gardens right along the Yarra River, containing a huge lawn, a rowing boat area on the Yarra and high trees. There are a few statues and a fountain as well as many flowers, as in the rest of The Domain. The Gardens continue onto the Queen Victoria Gardens, with a similar landscape and a few peaceful lakes. Further south is the Kings Domain with the Sidney Meyer Music Bowl, an open-air summer concert venue. In the southwest of the Kings Domain is the ugly pyramidal Shrine of Remembrance, a gray WWI war memorial, undeservedly touted as Melbourne's most famous landmark by some. To the east of the Shrine are the beautiful Royal Botanic Gardens, with a large collection of Australian and world-wide flora plants, including rainforest plants, eucalyptus trees, cacti and roses.
After visiting the gardens, we crossed the River Yarra on the Swan bridge, just north of the Botanic Gardens. This will bring you to Melbourne's sports heart, where you can find the Rod Laver arena, world famous for the Australian Open, the multi-functional Vodafone arena and the 1956 Olympic Park with its own olympic swimming pool, running lane, etc. A walking trail with some nice views on the city's CBD continues along the Yarra River back north to end where we started at the Flinders Street station.
The total distance of this walk is around 5km and easily fills a whole afternoon.