This paper has a methodological character, where we present a comprehensive formalism for constructing conserved quantities in the Teleparallel Equivalent of General Relativity (TEGR) and Symmetric Teleparallel Equivalent of General Relativity (STEGR). It was developed in a series of our earlier studies. and here, we combine them into a complete form. By employing the Noether method within a tensor formalism, conserved currents, superpotentials, and charges are constructed. These are shown to be covariant under coordinate transformations and local Lorentz rotations in TEGR, while, in STEGR, they are covariant under coordinate transformations. The teleparallel (flat) connections in both theories are defined using the 'turning off gravity' principle. Uniting such defined flat connections with the tetrad in TEGR and metric in STEGR, a new notion — 'gauge' — fruitful in applications, is introduced. The choice of various initial tetrads in TEGR or initial coordinates in STEGR leads to different gauges, giving different conserved quantities. Finally, we discuss an appropriate choice of gauges from a possible set of them.