To be honest, I'm not a great fan of using older texts for a beginner. I've seen texts written by an American learner of German once that sounded very old-fashioned. It turned out that he had learnt German primarily from 19th-century literature. Imagine somebody writing emails in English in the style of Charles Dickens or Jane Austen, only with frequent grammatical errors!
As for the text on the website (I only had a quick look at the first page) - it seems to be correct apart from one spelling error. However, parts of the vocabulary are dated and some of the words wouldn't be particularly helpful in everday life. By dated I mean that the words are comprehensible, but that people wouldn't speak or write like that in real life today. E.g.
Des Kaisers neue Kleider = dated/poetic genitive construction. Normal would be "Die neuen Kleider des Kaisers".
"recht geputzt" = dated in the sense in which the phrase is used in the text
"Rock" = dated in the sense in which the word is used in the text.
"es [ging] sehr munter her" = sounds slightly strange to me. I would say "zu" instead of "her", but maybe that's a regional thing.
"Zeug" = dated in the sense in which the word is used here
"dem Zeuge" = adding an -e to the word in the dative singular sounds old-fashioned (see also "dem Stuhle", etc.)
Handgeld = dated
What about easy readers in German? I read a famous work of German literature, Goethe's Faust, as a much simplified and modernized easy reader version with a Canadian friend. I think she enjoyed it a lot (I certainly did :)).
Have a look here, for example:
http://www.amazon.de/s/ref=nb_sb_noss_1?__mk_de_DE=%C3%85M%C3%85Z%C3%95%C3%91&url=search-alias%3Daps&field-keywords=daf%20lekt%C3%BCre
There are also "easy reader" versions of fairy tales which use only up-to-date colloquial German. E.g.
http://www.amazon.de/Rumpelstilzchen-nacherz%C3%A4hlt-Specht-Deutsch-Fremdsprache-Leseheft/dp/319301673X/ref=sr_1_18?ie=UTF8&qid=1367355241&sr=8-18&keywords=daf+lekt%C3%BCre
Maybe you can also find these books on Amazon.com. Otherwise, you could order them via Amazon.de, see:
http://www.amazon.de/gp/help/customer/display.html/ref=hp_gw_enhp/?nodeId=505532
On many of these books you will find terms such as "A1" or "B2", etc. These are European terms to express different levels of language proficiency. See here for an explanation:
http://en.wikipedia.org/wiki/Common_European_Framework_of_Reference_for_Languages#Common_reference_levels
To determine your present level, you could take a free online German test, e.g. here:
http://sprachtest.cornelsen.de/html/en_startseite.html