Your guitar may need to be "set up" properly. This means getting it into its best playing condition, and in particular, it includes adjusting the height of the strings above the fretboard (called the "action"). If the action is too high, the strings will be difficult to press down, just as you are experiencing. If the action is too low, the strings will buzz against the fretboard when you play—you don't want that, either.
You can learn how to adjust the action of your guitar yourself, but I highly recommend taking it to a professional repair shop for best results. A set up will cost around $30, and it will make a world of difference to how comfortable your guitar is to play.
Your guitar is like a bow (as in bow-and-arrow). The neck is a long, straight piece of wood with a very tight string attached at both ends, bending it into an arc. Obviously it's not as dramatic an arc as with a bow, but it's real nevertheless. Imagine for a moment that the arc really was as big as with a bow (meaning it would have extremely high action), and imagine trying to press the bowstring down to the bow itself as if you were playing it—that would be really, really difficult.
To help support the neck of the guitar, the neck has a steel rod inside it, called a "truss rod". This truss rod counteracts the string tension and keeps the neck from bowing too much. Because different kinds of strings exert different amounts of tension, the truss rod is adjustable, and it can be adjusted so that your guitar's neck has just the right amount of bow for your particular strings.
When you take your guitar to a professional repair person, one of the things s/he will do is adjust the truss rod until your action is just right. Different people prefer different amounts of action, so tell your repair person that you find playing chords to be uncomfortable, and s/he will make sure your action is low enough for playing chords without being too low.