You are not the only beginner here.
I started practising piano last month when I bought the e-piano Kawai CA5. Before that I never touched a piano key or played any musical instrument. I am a computer programmer, 33 years old. Since I dont have a teacher I need good books to teach me. I bought several books but none of them offered interesting music to practise. At last I found "The Piano Handbook" by Carl Humphries. It is a super book, I have a lot of fun with it. I also read the book by Chang, this book is great, it has a lot of useful ideas.
My goal is to be independent from music sheets when playing so I memorize everything I practise, a new piece may take me several days to memorize. I learn bar by bar, easy bars take me several minutes, difficult bars several hours :-) After I have the whole piece in my mind, first hands separate then hands together, I concentrate on playing it accurately, first slowly then at normal speed. I always record when I play so that I can quickly detect my weakness by listening to my record. By the way, you can improve your playing a lot when you know that your music sounds like sh... :-)
I try to practise piano everyday when I come home from my work. Sometimes when I come too late, I play the learned pieces for about 20 minutes before going to bed. When practising I divide my time in several short sections, every section lasts about 30 minutes. Small breaks between the sections are important, I always have the feeling your brain really learns the new notes when you rest. To improve the fingers flexibility I do the exercises by Isidor Philipp, they are very effective, at least for me (opinions on technical exerices are always conflicting).
If you are a beginner at piano, it is great because you decided to learn something great. Every pianist started playing piano as a beginner. I am not very concerned about when I will be able to play difficult pieces, may be one year, may be five years. What I know is that if you work diligently you will have little progress everyday, isn't it fun? :-)
Regarding Fuer Elise, I can play the easy part of it pretty well, two hands and without any music sheet.
Success to all of you in learning to play piano