An "Across the Course" Highpower Rifle match has two categories as stated above. There's Service rifle (M1, M14, AR) and there's Match Rifle. If it doesn't fall under the service rifle rules, it's a Match rifle. Generally, a service rifle can have internal modifications to improve accuracy (glass bedding, freefloated barrel, NM sights, etc). It's all done with iron sights. You can usually shoot F-Class (any rifle, any sight) at a match, but your score won't count and you'll look a little out-of-place (they're seen as a little bit of a novelty). Highpower shooters are generally fairly proud of using iron sights.
Across the Course bolt guns get fairly pricy (the sights alone are $pendy). They also have to be able to shoot the rapidfire stages, so that means they need to be milled to accept stripper clips (I've seen a couple bolt guns with detachable mags, but not too many--they're kinda new on the scene).
Shooting starts with offhand at 200yds slowfire.
Then you'll do sitting rapidfire. You start standing up. Once the targets come up, you have 60 seconds to sit down, shoot 2, reload, and shoot 8.
Then you move back to the 300yd line and shoot prone rapid. Same drill as previously.
Finally, you shoot prone slowfire at 600yds. Some places obviously don't have the full 600yds, so they shoot the course on reduced targets. All the same fun, but you don't have to deal with the wind like you would at 600.
The AR does just fine against the wind. It's the cheapest to get into match condition and the cheapest to feed, it shoots a little better and is easier to shoot well. You'll need something 75grs or more for 600yds.
Most CMP clubs have club rifles that you can borrow for the day. I'd do that for a little bit before sinking money into a heavy, expensive, and somewhat specialized rifle.
There are also long range matches that are fired prone starting at 600yds and finishing at 1000yds. Iron sights again, still shooting under the NRA Highpower rules.
If you want to shoot a rifle with a scope and bipod and those things, there are tactical matches held at some clubs. They use whatever local rules they have put together. To shoot long range (600yds+) with a scope, you'll need to spend a lot of money on a long range scope and mounts for it to be able to get enough elevation adjustment. A normal scope won't cut it.
Ty