There's bicycle paths/ways about Melbourne City Jade that can take you along the yarra and also the bay/beach front from Port Melbourne, maybe as far as Brighton Beach amongst other places.
Luna Park is a bit ancient and don't know whether there has been much in the way of renovations in recent years, but greta weekend market adjacent on beach side, and as beaches go, Brighton Beach does not rate too highly.
As for biking/hiking, the Yarra bikeway will probably take you out through the Yarra river corridor that links a number of larger green zones, Yarra Bend Park being first of those on fringe of the city.
You'll also find that the eastern side of the CBD has its own green zone centred about the Yarra river and Sports Centres precinct, and you can have a days nice walk without really leaving the city far behind you.
From Lygon street head a couple of blocks east to Exhibition Gardens, Melbourne and Indigenous museums there along with the old Royal Exhibition Buildings built along lines of a famous London hall from memory, wander through the gardens and own spring street a couple of blocks past Princess Theatre, Windsor Hotel and Parliament with the Melbourne Museum now occupying what might have been the old treasury building next door.
You are then at NW corner of the Treasury Gardens and head on the diagonal path and across the road is the Fitzroy Gardens that contains Captain Cooks Cottage and also a Conservatory (a little garden nursery actually) that are in the SW corner of FG.
Fitzroy Gardens are quite large in comparison to Treasury Gardens, but even more awaits - the Conservatory btw is well worth entering as they have some interesting phamplets on outdoor locations that can be of hiking interest amongst other good stuff not always available at all touristy info centres.
Back to the walk, and from the SE corner of FG it is but a short leap across Wellington Parade and a walk into the Melbourne Cricket Ground (world famous in itself, and has been home to some huge tours, Paul McCartney being one) reserve/gardens that doubles as a huge carpark when cricket and football games are on.
The MCG has a sporting museum if of interest, and an overhead walkway will take you across what remains of rail yards to the Tennis precinct, the main Rod Laver arena also being home to shows, Sher atop a battleship gun barrel being spectacular, and even things as diverse as motocross and big fukka wheeled trucks etc.
From there it's across the Yarra and you are at the Royal Botanical Gardens, arguably one of the best in the world.
On leaving the botanical gardens, there's the Kings Domain, Queen Victoria and Alexandra gardens that you can wander through getting back to the Yarra.
Head west (down stream) and you will find an abundance of bars/restaurants/food mall at Southbank for any hunger and thirst that has developed, the Casino if you wish to try your luck, another food mall, more bars and night clubs etc.
There's a footbridge back across the Yarra into the CBD.
There's a free city circle tram that does a city circuit and you could catch that in Latrobe St, just keeping walking into the city down Lygon street a few blocks and you'll come to Latrobe.
Lygon Street itself is a good location, it being referred to as Little Italy for Melbourne, there being plenty of restaurants and cafes for Coffee and delightful Pastries.
Further east from the exhibition gardens I mentioned earlier you find the area of Fitzroy, Smith, Johnston and Brunswick being main street names and that area has a lot of cafes, theatre restaurants and bars with live gigs.
To get some serious hiking in, Wilsons Promontary is a great location and
www.bunyiptours.com can get you there; you can also access
via this siteIf you're after something less expensive and closer to home, I remember reading somewhere that you could take a train to Lilydale (with your bike) and you can then get transported to Healesville where train used to go to, to then ride back to Lilydale - takes you through some of the picturesque Yarra Valley which is a big wine growing area these days.
Alternately, other than for ideas you might pick up from the Conservatory, I would suggest that if you caught a train out to Ferntree Gully (without bike) you will find some pleasant trails through the Dandenong Ranges NP.
Go about three stations further on and you will find
Sherbrooke Forest a treat and maybe even enjoy a
Puffing Billy outing; no not the name of a band or at least not one that I know of but can't help you there.
Another bootie from your neck of the woods, nina9702 is also currently in Melbourne, so maybe she'll know.
Have fun, and yes Melbourtne may not have a grand harbour but is still a great easy living city.