My partner and I are moving to Melbourne next year. What's a nice suburb close to the city?

Hi - you've picked some tough criteria there! Trams, mountains, bars and good food, and a city lifestyle.
Box hill is a great suburb in the east of the city which has great access to trams, trains, good food (amazing Chinese and Vietnamese) and a city type lifestyle. Box hill is also identified in Melbourne's planning scheme as a satellite CBD so there'll soon be high density living there with a high proportion of apartments.
If you like the mountains, then Belgrave to Upwey area is great with a good train line, good village atmosphere and a couple of cool bars and excellent restaurants further up in up in the hills.
If you like more inner city (skyscrapers) then look more to fitzroy or Carlton, Richmond or perhaps camberwell or even Doncaster which is getting some good attention. You'll get better access to trams and good bars and food in these suburbs.

Welcome to Melbourne - almost.
Inner city suburbs for your age group with tram access as follows, Brunswick, Fairfield, Clifton Hill, Collingwood, Fitzroy, Richmond, Abbotsford, South Yarra, Prahran, St Kilda, Port Melb, South Melb, Docklands, North Melb. Melbourne is Flat, we have no mountains other than the Lillydale-Sassafras region (no snow there). Closest mountains that get snow is Mt Bulla (I Think) where you can ski, 3 hours drive from CBD. Melbourne has the best restaurants in the world, so enjoy.