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We present the results of our investigations of physical properties for the novel Ce$_{1-x}$Pr$_x$CoGe$_3$ system performed with a number of experimental methods: magnetic susceptibility, specific heat, electrical resistivity, magnetoresistance, and thermoelectric power. Moreover, the electronic structure was studied by means of photoelectron spectroscopy measurements and first-principles calculations. All investigated compositions of the Ce$_{1-x}$Pr$_x$CoGe$_3$ series crystallize in the tetragonal BaNiSn$_3$-type structure. The lattice parameters and unit cell volumes decrease with increasing Pr concentration. On the basis of the measurements taken, a preliminary magnetic phase diagram was created. A continuous suppression of the long-range magnetic ordering was observed with increase of Pr concentration. The critical Pr concentration for magnetic moment ordering was determined from linear extrapolation of the ordering temperature $versus$ $x$ to the lowest temperatures ($T = 0$ K) and is equal to about 0.66. Based on the first-principles calculations we show how the substitution of Pr for Ce affects the electronic structure and magnetic properties of the considered alloys. Within a single model we take into account the magnetic ordering, fully-relativistic effects, and Hubbard U repulsion on Ce and Pr. The impact of Hubbard U on the results of calculations is also discussed. We present the valence-band analysis, Mulliken electronic population analysis, and calculated electronic specific heat coefficients. For CeCoGe$_3$ it is found that the $++--$ configuration of magnetic moments on Ce is slightly more stable than the $+-+-$ one, and also that the calculated value of total magnetic moment on Ce (including spin and orbital parts) is in good agreement with the measurements.