For now, I'll just mention a couple of common changes I've seen. I think these will probably pertain more to 関西 dialects.
One thing you may notice is the change that occur in vowels at the end of い-adjectives.
For example: 高い -> たけえ
つまらない -> つまんねえ
Here the -あい at the end turn into a long え sound. This is pretty regular and not difficult to apply. When extending this idea to other adjectives, I ran into a few trouble spots at first. It occured to me that this doesn't work for words like 暑い.
Here, you see a variation on this theme.
Examples: 暑い -> あっちい
寒い -> さみい
So here it's the same kind of thing. There's probably more to be said, but I'll save it for next time. I need some time to think over all this before I post anything unified or authoritative.
The ません->へん thing was mentioned above, but I thought I'd throw out a few more things concerning verbs. There's another way of changing the negative ending that I hear pretty much every day.
Example: 分からない 分からん
Here, aside from the special form with する （くる is sometimes きひん, but I personally don't hear it often) all that's going on here is ない->ん. Now this may not seem like a big deal, but the thing that originally struck me about this was how much it sounds like the negative as you often see it in literature (知らぬ for example).
One last thing I'll throw in for now is the copula. Everyone learns だ, but you will also hear や and in some contexts じゃ as well.
だ や じゃ
だろう やろう じゃろう
じゃない やん じゃん
and other such forms are possible.
Once I have the time, I'm going to flesh this out and post a big write-up of all the things I've picked up on. I also want to add that all this stuff is based on my observations and therefore is not intended to be an authoritative source on any of the topics I might decide to post. I merely have a strong interest in this sort of thing and I would like to post my observations. Look for something bigger, better, and more well-planned in the future.