First-Time Guitar Player Questions?

A few days ago we found amongst my late great-grandfather's things an old acoustic guitar, around 26 years old. I had it repaired and cleaned, and I have been attempting to play it, but I'm finding it rather difficult, and I just wanted to get some advice. For one thing, I'm finding it difficult to press down the strings completely, giving me a dull sound rather than a sharp tone. For another thing, it's incredibly painful to do so. Are these just problems that will eventually work themselves out as I play more? I imagine that the pain problem is due to my fingertips not having the calluses necessary for playing. As for the strings, is that due to the tension of the strings, or just my lack of experience? Also, I'm finding it to be difficult to press down one string without also pressing down the strings next to it. Is this also due to my lack of experience, or do I need a guitar with a different neck?