Is there an online resource (aside from random YouTube searching) that allows you to hear the various tunes for prayers/songs? For example, there are many to lecha dodi. I only know of one for eishes chayel and the best I could find was this youtube video which isn't how I remember. There are countless Shabbat zemiros and I want to learn some new tunes. From my last trip to Israel I heard many tunes which I wish I could remember.
As someone who descends from a long line of internet sleuths, here are some general (yet entirely specific) tips and suggestions:
1) First off, it is indisputable that piyut.org.il is to all appearances the #1 resource for traditional Jewish song. I should point out though, that as an ashkenazi, I have not really utilized it for researching ashkenazi neginah (with which I'm already familiar) but rather sefardi neginah (with which I'm culturally unfamiliar), so I'm not in the position to objectively evaluate its accuracy and thoroughness. Still, it clearly is fantastic.
2) For some reason, a lot of people I know don't figure this out on their own: Hebrew works in Google and Youtube. Seriously. No obscure third party plug-in is necessarily; just pull up google/youtube, hit alt-shift, and prepare for all your Jewish needs to be met. In our instance, this is what you'd get on youtube; the piyut.org.il page is the third result in the google search results. If you don't know Hebrew yet, this might a good reason to learn it!
3) Spotify has recently made its entire catalog accessible inside the browser, no longer necessitating that anyone download their software. At the price of listening to a few short commercials every now and then, you can enjoy all their music for free. If you have an account, you'd find 70 relevant tracks using their search.
Now to go compose a zemer thanking God for the internet...
For yekkishe (German etc.) nigunim of prayers, go to http://www.kayj.net/en/nusach.
Offtonic happens to have a whole lot of them. I strongly recommend it. (Full disclosure: it's my site, y'all!)
Another good source is the Zemirot Database
It provides recordings along with the words and a translation.
The website of Hassidut Modzitz (in Hebrew) has many interesting recording of their niggunim. Modzitz Hassidim are actually known as one of the dynasties that give the greatest importance to music.