You need:
Oil - manufacturer's recommendation is best. Your local service station will be able to help you; a socket set; an oil pan; a new oil filter; a dropcloth or newspaper.
Step by step:
Carefully lift your car using either jacks or ramps, ensuring to leave the car in park/gear and the handbrake on.
Turn up the heat:
Start your engine and bring it up to normal operating temperature - when it's warm, oil is thinner and will make draining the old oil easier, and when it's cold it picks up dirt particles and settles at the bottom of the sump. Running the car also means it gets more muck out. Once it's warm, shut the engine off.
Pan out:
Place your oil pan underneath the engine where you see a flat metal piece with a plug in it. This is the oil plug, which will need to be removed to drain what's already there. Carefully loosen and remove it, putting the plug safely aside. Wait until the oil has completely drained before screwing the plug back in, taking care not to crossthread or overtighten it.
Replace filter:
Oil filters could be on the side, front or rear of the engine. Unscrew it carefully - there are filter wrenches that make this easier if brute strength fails. Ensure the old filter's rubber gasket ring comes off at the same time - if it stays there the new one will leak.
Carefully screw on the new filter in a clockwise direction, again taking care not to crossthread. Tighten as much as possible.
Adding new oil:
Place the edge of the oil bottle against the open cap or over a large funnel, allowing the oil to pour slowly in without bubbling, adding the amount recommended by the manufacturer, and replace the oil cap.
Start engine:
Start the car and check if there is any leakage around either the oil plug or the filter. If it's leaking it's likely one of them is not tight enough.
Additional help
The head gasket sits between the engine block and the cylinder head and prevents coolant and oil from mixing by separately sealing oil and coolant passages. If the gasket is broken you may find oil seeping into the coolant - you'd be wise to leave this one to the professionals, and speak to your mechanic.