Whenever I try to play music using the Music app on my iPhone 4S (iOS 6.1.3) the phone skips over 30-40 songs before it begins playing one. If I try to skip the current song it jumps ahead another 30-40 songs again. If I try to play one album it will skip through all the songs in a second or two and not play anything at all
When songs are grayed out and the song won't play, it may be that the song did not completely transfer over to your iPhone. You can resync content to the iPhone by unchecking "Sync Music" and then clicking "Apply" and then checking "Sync Music" and then clicking "Apply" again.
You can find the syncing steps in the following article:
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