Installing a new ceiling light is easier than it sounds. Once you remove the old light (taking the proper precaution of cutting power at the circuit box, not just at the wall switch), it's just a matter of rewiring the new fixture in the same way the old one was wired. One mistake people sometimes make is to install a smaller fixture, which often leaves a ring of unpainted ceiling that was under old fixture. If you don't want to repaint, just make sure the new fixture is at least as big at the base as the old one.
Kill power to the circuit, turning it off at the house circuit box. Climb a step ladder to the existing fixture. Take the fixture down from the ceiling by unscrewing the mounting screws that are holding it there, pulling the fixture down to expose the wiring, then disconnecting the wires. You'll be left with an open ceiling electrical box with three wires (white, black and bare copper) hanging down from it.
Hold the new fixture near the electrical box. Use a wire cap to connect the black wire from the box to the black wire from the fixture. Connect them by holding bare ends of the two wires side by side and twisting the cap over them both at once. Wrap the cap in electrician's tape.
Repeat the process to connect the white wire from the box to the white wire from the fixture.
Wrap the end of the bare copper grounding from the box around the green grounding screw in the fixture. Tighten the grounding screw.
Hold the fixture to the electrical box, pushing the wire into the box as you do. Mount the fixture with the provided screws, securing them through the holes in the face of the electrical box. Restore the power.
Always kill power to the circuit before working on any fixture.