As many people do I often spend long hours in front of the computer. This can be tough on the eyes if one is not careful. During my 20 years of computer use I worked out a series of habits and techniques that help minimize that strain. I'm sure you will also find them helpful:
Low contrast work environment. I light my workspace with a 2 x 60W lamps so my workspace is evenly lit and the contrast between the monitor and the rest of the room is minimal. I position the lamps in such a way so each pans directly at the wall it stands next to and is placed slightly behind the line of the monitor. The light bounces off the walls and this produces a nice soft ambient illumination. I found that the lower the contrast between the room and the monitor the less strain is put on the eyes. But, you can try the opposite and sit in a dark room and look onto a bright PC display. You will get quickly tired in your eyes.
Refresh rate. This refers to how often the screen is being updated. I found the best refresh rate to be between 85 Hz and 120 Hz on CRT monitors. If your screen is flickering, it's probably due to low refresh rate. You can adjust this by right clicking on the desktop and locating the appropriate tab.
High quality cables. Low quality cables that connect your computer and the screen can cause massive interference like vertical or horizontal bars, blurred fonts, and similar. So if you are having such problems replacing the cables might be an easy solution (no need to buy a new monitor yet). Finally, as each additional connector degenerates the signal don't use extension cords. Use a longer cable instead.
Use quality monitor and graphic card. Don't buy cheap clones or other low cost solutions. They don't last long and often have problems with higher refresh rates at higher resolutions. It's better to invest in something more expensive that will last you much longer and provide better functionality.
Take frequent breaks. I take mine every 30 minutes. A few minutes off the screen is usually sufficient for shorter breaks. Actually, my daily routine is structured in such a way that I get 2 - 3 longer breaks each lasting 30 - 45 minutes. These are when I cook and eat food, which is a natural break every 3 hours or so.
Relax your eyes. There are many ways to do that. I developed a simple and effective method that consists of two steps. First gently massage your muscles around the eye sockets, including the area just around the eye itself. This typically takes me up to a couple of minutes. Then I sit back (or lie down) in a relaxed position and put a piece of dark blanket (or whatever will work) over my face so no light can pass through. I close my eyes and consciously relax my eyes and muscles in my whole body. This is great for eliminating muscle tension and after about 15 minutes my eyes feel refreshed and relaxed.
Lower the screen contrast. If you work with graphics or frequently view images onscreen you should properly adjust your gamma settings. This is important. Otherwise the images you see may look too dark or too bright, and the colors may not be as they were intended to be seen. However, if you primarily use your computer for text based activities, like reading and writing, you can lower the monitor brightness/contrast and reduce the monitor color temperature to around 6000. I have two main computers. One for art and the other for doing text based activities (email, internet, programming, research, etc), and the difference is big. You can adjust these through the front panel of your monitor.
Adjust your onscreen working environment. Basically, I adjust everything I see on screen to be easy on the eyes. Window layout style, desktop background, fonts (color, type and size), background/foreground colors, etc. Antialiased black fonts on a light grey background works best for me.
Stay healthy. There is no substitute for that. Eating a healthy diet that consists of natural and minimally processed or whole foods, regular exercise and a good nights sleep is essential for proper body function.
Finally, there are the very obvious ones like avoiding reflections and glares on the screen, minimize repetitive computer work (automate as much as you can), and avoid fluorescent lights.
All these simple adjustments can help keep your eyes rested. I think if you try any of these you will notice an improvement. Try one at a time to see how they work for you.