SpaceX is going full steam ahead with Big Falcon Rocket (BFR).
Musk thinks that the booster part (currently called "Super Heavy") will be the easier part. Therefore, SpaceX is currently focusing on the upper stage (currently called "Starship" and before that Big Falcon Spaceship or BFS).
For those losing track of Musk's repeated renaming of the project Wikipedia provides up-to date section on BFR nomenclature.
About renaming BFR to Starship, Musk's has said:
Technically, two parts: Starship is the spaceship/upper stage & Super Heavy is the rocket booster needed to escape Earth’s deep gravity well (not needed for other planets or moons)
All rockets capable of reaching Earth's orbit with useful payload are built with at least two stages. Sometimes more. This is due to Rocket Equation. Dumping the dead weight of an empty first stage allows for more efficient use of energy. Super Heavy Starship (or BFR) is also built with two stages: the first stage is called "Super Heavy" and the second stage is called "Starship" (or BFS).
An interesting aspect of the Starship is that it will be able to fly without the Super Heavy. That will allow it to return from other planets and moons to the Earth. It will also be capable of single stage to orbit launch from the Earth without any useful payload. The capability is intended to be used for testing of entering planets atmospheres from super orbital velocities like Mars/Moon transfer velocities. According to Musk there are certain heating parameters that scale to the eighth power with regard to speed.
I will do a full technical presentation of Starship after the test vehicle we’re building in Texas flies, so hopefully March/April
Previously the Starship was supposed to be built using composite materials, but the current design is using stainless steel.
Stainless steel is correct, but different mixture of alloys & new architecture. Unlike Atlas, Starship is buckling stable on launchpad even when unpressurized.
Will SpaceX start testing a Starship (BFS) before 2020?
The question will resolve positively if we get information from SpaceX in the form of a video or a photo showing that any early test version of Starship/BFS (the second or upper stage) has been launched before 1 January 2020. Launched here just means that all legs must be unambiguously above the ground when the vehicle is still in its original shape. After that, it could even explode without affecting how this question resolves.
The question will resolve negatively if no test is attempted or if the Starship will not manage to get off the ground in one piece before 2020. Several attempts are allowed.
Related question: Will SpaceX test-launch the BFR before 2025?