schmange has a >video lesson< posted to show to change the strings. He takes all of the old strings off at once, before installing the new strings. I pprefer to change strings one at a time, so that I can tune the new string to teh ones that are on the guitar as I go. If you have a good tuner handy, this is not so much of an issue. Some guitar bridges are held on only by the string tension, so you need to keep some strings in place, or the bridge falls off. You can make up your own mind on how you want to do this.

You can try out a different guage of strings without doing any damage. Things to watch for:

• With heavier strings, there will be more tension on the neck, so it will bow a little more. This will cause to the strings to be higher above the frets than the lighter set was. This won't damage anything, but it is a bit harder to play, so if you decide you like the heavier strings, you may want to get the truss rod adjusted by a tech.
• The slots in the nut were probably cut for the guage of strings that were on there. So the thicker strings will not fit quite right. They may bind in the slots a bit, making tuning a bit tricky. They may also sit a bit higher above the frets, because they won't go the bottom of the slots. This is something else a guitar tech can fix for you.
• The tone is likely to be richer, and more full. They will also sound louder.

So go ahead and try the next size up from what you have now. If you prefer the way they sound and feel, take your guitar in to have it set up for the heavier strings. This will restore the action and adjust the intonation so that it tunes more accurately.