When you have the correct oil begin by starting the mower’s engine, once its nice and warm stop it and disconnect the mower’s spark plug. Warming the engine will help the oil to pick up all of the little particles that have collected in it and disconnecting the spark plug will ensure the mower will not power up during the oil change. Wipe the area around the oil plug to prevent additional debris getting in when the cap is opened. Drain the oil by tilting the mower and emptying it into a jug, you could also use a oil removal pump/kit.
There are oil change kits which include a pump, hose and collection can. If your mower has an oil filter, this should also be replaced at the same time.
Most handy kits also include oil and filters, available from garden machinery dealers, Homebase and some garden centres.