How can I learn to sing/chant Hindu or Tibetan mantras?

Study them.

Ram Dass has some audio CDs out, you can search them out on Amazon.com

Go to your local Hindu temple.

Google ……………you tube

Maybe you could search for some mp3s and try to imitate them?

Find someone to teach you, or perhaps a website or audio recording.
Then practice, practice, practice.

Also look in the youtube.com, prapatti.com.
There would be too many of them, choose the type of mantra you would like to chant and go from there.

There are several places on the Internet where you can purchase CDs of chants.
http://www.amazon.com (do a search Buddhist Chants Mantras do a separate search Hindu Chants Mantras — there are frequently snippets you can access to hear the quality)
http://www.namsebangdzo.com/ (use “Tibetan Chants & Music” from the sidebar)
http://www.fpmt.org/shop/ (select “Audio & Video” from the sidebar and then select “CDs & CD ROM”)
http://www.snowlionpub.com (select “Music & Chants” from the sidebar and then the category you’re interested in from the pop-up)
http://www.dzogchen.org/chant/index.htm (has some selections in Real Audio format you may download and may purchase CDs — not too pleased with his chanting in all honesty but that’s a personal opinion)
You can also download free MP3 format chants and mantras and then burn your own CDs by using sites from
http://www.emusic.com/album/Buddhist-Monks-of-Maitri-Vihar-Monastery-Tibetan-Mantras-Chants-MP3-Download/10595287.html
http://www.buddhanet.net/audio-chant.htm
http://www.buddhanet.net/audio-library.htm (has some meditation instructions as well)
I’m sure there are others if you google Buddhist, Mantras, Chants, mp3 or a similar search.
I hope this helps.
May all be at peace.
John

At the feet of a music teacher. Second best, obtain CDs.

You can learn them by listening to some of the audio CDs, now available in plenty or down loading from Music India Times.