The space shuttle does have rockets attached to it! It has three main engines (SSME). The large fuel tank contains the liquid oxygen and liquid hydrogen that is pumped through piping between the large tanks and Space Shuttle. The booster rockets (SRBs) provide the propulsion for the large tank. The SRBs then separate when the shuttle gets high enough to atmospheric drag reduces to the point that the tank can travel with the shuttle without being torn away.
Nuclear thermal propulsion alleviates the need for the chemical reaction between hydrogen and oxygen; the hydrogen is simply passed through the reactor core and thermal energy is conducted from the reactor fuel elements into the hydrogen. Even with only hydrogen, a Space Shuttle would still require a tank (vessel) to hold the hydrogen, whether external or internal.
The space shuttle is design to return to earth much like an aircraft, and thus serves in low earth orbit. For venturing further out from LEO, one would require a different configuration, much like Apollo system or the successor, Artemis.
https://www.nasa.gov/humans-in-space/artemis/
https://en.wikipedia.org/wiki/Artemis_program
So far, only ground tests of NTR systems have been tested (NERVA). There have been no in situ deployments of NTRs, so it is not clear how effective such a system would be. Theoretically, the specific impulse of a nuclear thermal system is about twice that of a hydrogen-oxygen chemical propulsion.