Tharwa is a great idea for a day out and there is a lot to see in a small area. You could consider riding there south through Canberra, or if you are keen then try the loop ride from Stromlo Forest Park south through the suburbs to Tharwa, where you cross the Murrumbidgee River and return to Stromlo Forest Park on the other side of the river via Tidbinbilla Nature Reserve and Cotter.
Tharwa and Hall were towns in NSW before the ACT even existed. The ACT was formed by purchasing three large farms. The Lanyon Homestead is now a museum a few kilometres north of the Tharwa Bridge.
The single-lane bridge that crosses the Murrumbidgee River to Tharwa is new but built in the style of the old wooden bridge it replaced. The ACT Government proposed a new modern two-lane bridge here but the Tharwa residents protested and demanded that it at least looked the same as the old one. The bridge was opened by locals with a party on the bridge.
Tharwa is on the Bicentennial National Trail and just a few kilometres north of the Namadgi National Park and visitors centre. The area has retained its traditional character. Expect rural road standards with narrow shoulders and no on-road or off-road bike infrastructure. You can ride on the management trails in the Namadgi National Park and the dirt roads in the Tharwa area (excluding the farms of course). Smiths Road is a tip for the gravel rider. Remember the Bicentennial National Trail is a good guide of where to go as it is publicly accessible and well documented on maps in the ACT including OpenStreetMap.
The Murrumbidgee River Corridor is currently under review by the ACT Government.