Steam Turbines had a long history, and rockets had a long history, so both turbine aircraft and jet aircraft would have been considered. Propeller aircraft and pulse-jets already existed.
Were people trying to build turbo-prop aircraft? Or was it an impractical idea that only became possible after the development of fan-jet engines?
The first turbojet aircraft to fly was the Heinkel He 178 on 27 August 1939. The first turboprop to fly was a Gloster Meteor (I.E. a jet model) which had been fitted with two “Rolls Royce RB.50 Trent” engines. These were actually converted Derwent II turbojet engines that had been fitted with five-blade propellers. Their first flight on this testbed was on 20 September 1945. Early turbofan prototypes, I.E. the kind of engine typically used on modern “jet” aircraft, were ground tested in both Germany and the UK in 1943, but the Rolls Royce Conway “bypass jet” engine became the first commercially-used turbofan in the 1950s. After Pratt and Whitney developed its JT3C/J57 turbojet engine into a turbofan, the JT3D/TF33 (first flown in 1959), turbofans really took off and became standard on commercial aircraft within a few years.
You should know that development of these different kinds of engines overlapped to a considerable degree and it wasn’t necessarily a perfectly linear development from one kind to another. The concept for the turboprop was first published in 1928 by Hungarian physicist and engineer György Jendrassik in 1928, who patented it the following year. Also in 1928, in the UK, Frank Whittle was advancing his idea for a turbojet, an idea which had first been patented in 1921 by the Frenchman Maxime Guillaume.