Framing could be an issue for two reasons: (1) width of stair kit and (2) structural support in the attic for weight load.
Some amount of re-framing or bracing is typical. Adding a cross-brace is in your range of skill if you can cut and drill. Here is a basic primer on doing the install without too many complications. The guy is very patient and detailed:
It's always fun to find "good old boys" who take the time to show you a more advanced version of the stair install. (Multiple videos) This guy is a lot of fun and his series of videos takes on the more complicated versions of the process. It should give you some confidence on dealing with crossing over ceiling joists.
The weight load is another point. As Neal says, DO NOT cut engineered trusses. You can add cross-braces to the joists for strength. However, I suggest NEVER putting heavy items in the attic. Keep it to light boxes.
Also, you will encounter ROMEX that runs your electrical all over the house. DO NOT put plywood over the wiring where it overlaps the ceiling joists ... this pinches the wiring and could cause a fire.