We provide and analyze a periodic Anderson model for studying magnetism and superconductivity in UTe2, a recently discovered candidate for a topological spin-triplet superconductor. The 24-band tight-binding model reproduces the band structure obtained from a DFT + U calculation consistent with an angle-resolved photoemission spectroscopy. The Coulomb interaction of f-electrons enhances Ising ferromagnetic fluctuation along the a-axis and stabilizes spin-triplet superconductivity of either B-3u or A(u) symmetry. When effects of pressure are taken into account in hopping integrals, the magnetic fluctuation changes to an antiferromagnetic one, and accordingly spin-singlet superconductivity of A(g) symmetry is stabilized. Based on the results, we propose pressure-temperature and magnetic field-temperature phase diagrams revealing multiple superconducting phases as well as an antiferromagnetic phase. In particular, a mixed-parity superconducting state with spontaneous inversion symmetry breaking is predicted.