Observing is more than just opening your eyes (although that certainly helps). It is an acquired skill. And while buying a big telescope will almost certainly help you to see dimmer objects, good observing practices can help nearly as much - and the cost is significantly lower. Presented here are a few good techniques to practice and master.
The first and easiest practice to adopt is called 'Dark Adaptation'. Your eyes are very sensitive - capable of seeing very faint objects. However, if you've just come out of lighted house or away from any white light source (like a lantern or bright flashlight), it will take your eyes some minutes before they are fully dialated and free from light 'imprints'. During this time of dark adaptation, you must avoid looking at bright light sources such as headlights, streetlights and even the moon. Some people close their eyes. Others look up at the stars in the night sky. Even a few minutes of this dark adaptation will help you see dim objects much better. However, it may take over 15 minutes for your eyes to fully adapt to the dark.
The second practice to adopt is to be patient when you observe. Astronomical observing is an exercise in noticing sutbleties. When you observe, take several minutes to really look at an object. As you look, you will notice more and more of the structure of your target. Atmostpheric turbulence (which is nearly always present) can temporarily subside revealing details that were invisible only a moment before. Some people will tap on their telescope lightly to introduce a tiny bit of motion - the motion can cause your eye to see very faint light that you might have otherwise overlooked.
Averted vision is another way to see the unseeable. Your eyes have two different types of photoreceptors, rods and cones. Cones give you more detailed color vision, but rods are more numerous and more sensitive. Rods really come into play more for your peripheral vision, so if you use that part of your vision, you can increase the sensitivity of your eyes significantly. The idea is to look in the eyepiece just a bit to the right (if you view through your left eye) or left (if you observe with your right eye) of your target. This will cause more your rods to look at the object. Often the object will suddenly 'jump out' and you'll be tempted to look straight at it. Avoid the urge and continue to use averted vision.
While you're observing, it's important to change the magnification of your telescope by switching eyepieces. Detail that may be present at one magnification may be lacking at another magnification. Often switching to a lower magnification will help your eyes see faint objects better than higher magnification. Other times, you can see more detail - especially on planets - by increasing your magnification as high as your telescope and the seeing conditions will allow. It's something that varies every night for every target. You just have to try several combinations and see what works.
Finally, remember the creature comforts. It's really hard to enjoy observing if your feet are freezing cold. Dress appropriately, so you can remain at the eyepiece for extended periods of time.
A more detailed guide (pdf format) for beginning observers has been prepared by Nils Allen, and may be downloaded/viewed by clicking here.