We synthesized strontium titanate SrTiO$_3$ (STO), Zr doped $text{Sr}_text{1-x}text{Zr}_text{x}text{Ti}text{O}_3$ and (Zr, Ni) co-doped $text{Sr}_text{1-x}text{Zr}_text{x}text{Ti}_text{1-y}text{Ni}_text{y}text{O}_3$ samples using solid state reaction technique to report their structural, electrical and magnetic properties. The cubic $Pm$-$3m$ phase of the synthesized samples has been confirmed using Rietveld analysis of the powder X-ray diffraction pattern. The grain size of the synthesized materials was reduced significantly due to Zr doping as well as (Zr, Ni) co-doping in STO. The chemical species of the samples were identified using energy-dispersive X-ray spectroscopy. We observed forbidden first order Raman scattering at 148, 547 and 797 cm$^{-1}$ which may indicate nominal loss of inversion symmetry in cubic STO. The absence of absorption at 500 cm$^{-1}$ and within 600-700 cm$^{-1}$ band in Fourier Transform Infrared spectra corroborates Zr and Ni as substitutional dopants in our samples. Due to 4% Zr doping in $text{Sr}_text{0.96}text{Zr}_text{0.04}text{Ti}text{O}_3$ sample dielectric constant, remnant electric polarization, remnant magnetization and coercivity were increased. Notably, in the case of 4% Zr and 10% Ni co-doping we have observed clearly the existence of both FE and FM hysteresis loops in $text{Sr}_{0.96}text{Zr}_{0.04}text{Ti}_{0.90}text{Ni}_{0.10}text{O}_3$ sample. In this co-doped sample, the remnant magnetization and coercivity were increased by $sim$1 and $sim$2 orders of magnitude respectively as compared to those of undoped STO. The coexistence of FE and FM orders in (Zr, Ni) co-doped STO might have the potential for interesting multiferroic applications.
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