Brisbane cops a lot of flak because it used to be a big country town (and sometimes still feels that way despite the skyscrapers).
The CBD has Queen Street for shopping, with City Hall around the corner (currently closed to repair the building from concrete cancer). If you go to the end of Albert Street, you'll find the Botanic Gardens and Qld Parliament House. The inner suburbs surrounding the CBD are the most interesting in terms of restaurants, pubs, bars, live music etc. These include:
Fortitude Valley: the main entertainment district and gay area centred around Brunswick St, Ann St and Wickham St. The main gay bar is The Wickham, gay club is The Beat, and The Sportmans Hotel ('Sporties') is in the next suburb, Spring Hill.
New Farm: Large park which is great for picnics, also has the Powerhouse which is a venue for exhibitions, concerts, live music. There's a restaurant and bar as well. Methyr Village has the New Farm Deli.
West End: a bit more multicultural due to the Greek, Lebanese and other immigrant communities' influence. This quarter has a high student population as well as alternative/progressive people. Thai, Korean, Vietnamese, Greek, Indian, Peruvian, Italian, Vegetarian and Vegan restaurants all within a stone's throw of each other. Main areas are on Melbourne, Boundary and Hardgrave Streets, with Saturday markets in Davies Park. The University of Queensland is directly across the river from here and you can get there on the ferry, called the CityCat, which will also give you the best tour of the CBD.
Southbank: The main tourist attraction (and hence, pricey). As well as the artificial beach, there are also weekend markets, cinemas, restaurants, a bouganvillea walk, river quay, maritime museum, and also a beautiful carved Nepalese pagoda. Opposite the parklands is the cultural centre with the Art Gallery, Gallery of Modern Art, State Library and Museum.
Kangaroo Point: as well as the Storey Bridge and associated hotel, there are the cliffs were you can do rock climbing, or you can walk along the top and get great views of the city.
Paddington/Milton: More restaurants and clubs around the old police Barracks and Caxton Street as well as up along Given Tce. Suncorp Stadium here if you want to see a football or rugby match.
Wooloongabba: the cricket stadium is here, and in winter it's used for AFL. Directly opposite are some great restaurants, one of which is Bistro Bistrot.
The Gap and Brisbane Forest: You can take a bus or otherwise drive out to the end of Waterworks Road to go bushwalking or gentle hiking around Enoggera Reservoir. Other outdoorsy places to see include Lone Park Sanctuary (for koalas), North Stradbroke Island (take the train to end of Cleveland line and a bus will take you to the ferry), the Glasshouse Mountains (especially Maleny), Samford Valley, Sunshine Coast around Caloundra, Maroochydore, Mooloolaba. You could also take the Shorncliffe train to Sandgate - this was where Brisbaners used to go when they went to the 'seaside' in the days before the Gold Coast was even a few shacks down in Burleigh Heads. There's a jetty here, plus good fish and chip places.
Of course, there's the Gold Coast further down, which is a long sprawling strip of towers. The main beach is Surfers Paradise and Broadbeach, but you may find Burleigh Heads more relaxing and local. You can then head around to Kirra Beach and Snapper Rocks.
That's about the most comprehensive guide I can give you off the top of my head ^^ Brisbane isn't Sydney or Melbourne, but hopefully you'll discover its own laid-back subtropical charm