Bus and Train it to the bush
Surrounding Brisbane are many fantastic hikes, however few of them are accessible for someone without a car. There are still a number of great hikes that you can access by public transport. Also it makes it more pleasant to sit back and relax on a bus or train rather than having to drive for hours.
Here are a few of my favourite public transport accessible hikes.
Mt Coot-ThaBeing close the city centre of Brisbane,
Mt Coot-tha is easily accessible by buses with numerous walks available and is a popular location for people looking for
training hikes as well as visitors to Brisbane.
Most people will take the bus to the summit and walk the summit trail, but there is a lot more you can do.

Summit Trail, Mt Coot-tha
There are several locations where buses will drop you off near the start of a hike. The 425 bus takes you to the popular Honeyeater track. This track takes you from the base of Mt Coot-tha to the summit up a trail that is not too steep, but is still a good work out. The same bus takes you further down to the start of the Kokoda and Bell Bird tracks, which lead to many additional tracks you can take.

Honeyeater Track up Mt Coot-tha
On the other side of the mountain, there is the 375 bus to Gordon Road in Bardon, where there are a few trails including the Malcalata Track that takes you to Simpsons Falls. Remember all these tracks link up, so you can start on one side of Mt Coot-tha easily and walk across to the other side.
Walkabout Creek and Enoggera Reservoir
Walkabout Creek has the lovely Green Tree Frog Cafe, Zoo, family friendly walks. One popular one is the Enoggera Resevoir Circuit, which unfortunately is not the circuit of the lake, but does take you along the lake.

Enoggera Resevoir
However the more adventurous can get on the fire trails that
circle the lake/LINK]. This is definitely the most beautiful hike in the Brisbane city limits.

Hikers doing the circuit of Lake Enoggera
Recent changes to the buses to the 385 buses to The Gap means that to get to walkabout creek you need to walk up to the road from the last bus station. Which is not much of an obstacle for a hiker.
This also the start of the D'Aguilar National Park and there are varies fire trails and unmarked bush tracks that experienced hikers can spend days on as they head up into the mountains.
North Stradbroke
North Stradbroke Island is one of Brisbane's great treasures. Most people take the ferry across and then take the bus to Point Lookout and do the gorges walk along the beaches and clifftops to Cylinder Beach.

Cliffs at North Stradbroke
There are other hikes available on the island, particularly Brown and Blue Lakes. Often better to go by car, but for a keen walker it is possible to walk along the road to the start of these hikes from Dunwich. Also consider hiring a bicycle from Dunwich and cycling across the island to main beach.

Main beach at North Stradbroke
Gold Coast Beach Hikes
The coastal strip of the Gold Coast is easily accessible by the many buses and the tram that runs up and down its length. Burleigh Heads National Park is an absolutely lovely location with circuit of about 3.6 km.

Walkers at Burleigh Heads
A longer option is to walk through the coastal bushlands at South Port to the spit. While South Port is known for its resorts and theme parks, half of it is still bushland. You can do an interesting circuit here going up through the bush and back along the beach. Also a great location for cycling. Out at The Spit is a nice little fish and chip shop. See this area before developers move in and turn it all into more theme parks and resorts.

South Port is just north of Surfer's Paridise
Karawatha Forest
Some of my friends consider this area amazing. The bushland reserve near Trinder Park Station in Logan provides some reasonably interesting hiking.

Karawatha Forest Park
If you hike on a Sunday morning remember to drop in for some South East Asian Street Food at the [LINK=https://www.weekendnotes.com/woodridge-markets/]Woodridge Train Station Markets.

Asian street food at Woodridge markets