I'm interested in learning Archaic English. As a starting point, I guess simple texts that are easy to comprehend would be a good choice. I would appreciate any suggestions.
There really isn't any "language" called Archaic English. Do you mean Old English? If so, there are textbooks for studying that. Look for books on Anglo-Saxon or Old English.
If you mean Middle English (spoken at the time of Chaucer), you can find texts on that as well.
Shakespeare's writing sounds archaic to our ear, but is actually an example of what is called Modern English.
Are you interested in archaisms in the modern language? If so, you might see if you can find a copy of The Archaicon: A Collection of Unusual Archaic English somewhere.
If none of those work for you, perhaps you can elaborate on what you're really looking for.
If you want to read something more recent ( what you might think of as the same language we use now but with those archaic usage such as thee and thou ) both Shakespeare and the King James Edition bible are influential and well written. It is also easy to find modern translations or adaptations of both so you can figure out any tricky phrases by comparison with a different version.