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It may be time to replace the grout and re-seal it. Call around and get an estimate for it to be professionally done. It's worth every cent. Ask the installer what he recommends for keeping it nice. There is that one automatic thing that hangs on the shower by a hook that sprays Scrubbing Bubbles stuff all around the shower to keep it clean. I hear it works really well, but I don't have one to swear by it.
The kind of soap you use in the shower may be a big culprit. Regular bar soap of almost any kind is going to leave scum and then mildew starts to grow in it. Try to use shower gels instead. They rinse away better, though it doesn't feel like it for some reason. Get a squeegee for th shower and get as much water off the tile after you shower as you can. Install and use a room heater or exhaust fan religiously to keep the humidity levels low enough to keep the mildew at bey.
Good luck.