Far and away the best source of Lieder and Song Texts is Emily Ezust's website. It "is an archive of texts used in 72,251 Lieder and other classical art songs (Kunstlieder, mélodies, canzoni, романсы, canciones, liederen, canções, sånger, laulua, písně, piosenki, etc.) as well as in many choral works and other types of classical vocal pieces. The collection currently indexes 43,824 texts and includes 9,255 translations to English, French, Italian, Dutch, German, Spanish, Portuguese, and other languages. The site has frequent updates." There were technical problems for a week, but it's back online.
Quite frankly, there's nothing like it. For the last 15 years, it's been an essential tool for song people, endlessly useful. Reach it HERE or use the link on the right of this page.
Obviously anyone with even the most basic interest in Lieder already has the texts of famous pieces. It's an oxymoron. It''s impossible "not" to have a recording of Winterreise or any of the great classics even if you aren't specially into song. Emily's site is wonderful because it goes far beyond "greatest hits" and covers thousands of songs whose texts aren't easily available. Also many composers and poets who aren't generally known.
But text alone isn't what song is about. The site has translations in several languages, as text is of limited value if meaning isn't known. Translations are a tool, not the "real thing", so it's usually a good idea to do your own, even if there's a very good translation. (get Richard Stokes's translations for the Gold Standard). Doing your own is good, because that gets you deeper into the text and the way it works. And the way words work with music tells a lot about the composer and his response to the text: which in turn tells you something about the composer's mindset. That's the joy of hearing different settings of the same text.
Often, you think you know something because it's familiar and take it for granted. It's not easy to blank out meaning in languages you know, so it's a good exercise to listen to songs in languages like Finnish or Czech. You grasp meaning by intonation and inflections in the voice. It's more intense than ordinary listening. Even if you miss out on the full import, you focus on the innate music in the sound of the language. That's why modern composers are fond of obscure languages, which convey something, but aren't explicit.
Once I went to a recital of a new cycle in ancient Sumerian or something like that. Huge audience, which was a big surprise. "Guess no one here speaks...." said the composer. Dozens of hands shot up. "Yes, we do!" The concert took place in Oxford, one of the few places in the world where you'd get a room-full of people who'd specially come because they knew the obscure language in question.
Perhaps someone reading this can advise, what is the status of translations from copyright texts? Usually you need formal approval, but would one way to get around this be to summarize in prose (not as direct translation)? There's one very good song cycle, where the composer wrote his own texts, but his English isn't nearly as good as his native German (or his music). Chances are the work will be "closed" to non-German speakers, which will be a pity, as it's an excellent cycle.