Making your bike lighter isn’t especially difficult. Be prepared to spend some money, though.
Concentrate on rotating weight first - wheels, pedals and so on. This will have the biggest impact on how light your bike feels when you're riding it.
Buy lighter tires. This is the most effective and cheapest way to bring down the weight of your bike.
Buy a lightweight wheel set. Look for something hand-built.
Buy lighter pedals. You may want to consider clipless pedals, which will also improve your comfort and efficiency.
Upgrade your fork. Replace your heavy stock suspension fork with a lighter, higher-performance model. Or replace your heavy steel road fork with a carbon fiber fork to decrease weight and smooth out road vibrations.
Consider new cranks or a lighter bottom bracket.
Buy a lightweight seat and seat post. Compared with some other upgrades, these are relatively inexpensive, and you'll feel the difference when climbing out of the saddle.
Consider titanium if you really want to go for the ultimate in lightweight parts. Titanium cassettes, skewers, bolt kits, bottom brackets, stems, bars, seat posts and so on are all on the market. Each will bring the weight down but will also make the price shoot up.