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You do not need long fingernails to play guitar. In fact, it's best to keep your fingernails short! As for the callouses, these may be unavoidable. If you would like to avoid them as much as possible, ask your guitar instructor about lowering the action of your guitar (the action is the distance between your strings and your frets) so that the frets are easier to press down on.
If you just want to learn a few chords to strum along to some songs, then learning to play the guitar really isn't that hard for most people. Many students are able to learn a few chords and get a decent strumming pattern going after a few weeks. If you've had prior experience playing an instrument, you can expect to pick it up after just a few lessons.
Now I know you might be confused by this diagram, so let's take a moment to explain it. First off, the top of the diagram corresponds to the nut on your guitar, that's the part where the strings end, farthest from your body. Left to right, the strings are 6,5,4,3,2,1. So as you look at the diagram, the string on the left is actually the 6th string on your guitar, the low E string.