edit: sit down and read through the manual, it will be very helpful.
The action is basically the bolt, bolt handle, and all of the parts contained inside, and the receiver that houses the bolt and serves as a mounting point for the barrel, there is usually a spring inside the bolt that is compressed upon cycling the bolt. This force is released on the firing pin which then contacts the primer on the casing.
The trigger group will be attached to the base of the action. The safety systems will vary between firearms. I would advise you to not take these things apart unless you know what you are doing.
With an unloaded bolt action. Check to make sure its unloaded again. Take the safety off, open the bolt, pull the trigger and keep it pressed while pulling the bolt and it should come straight out the back of the rifle. You can get a look at how it works. Some bolts can be disassembled easily with the help of certain tools, be warned that if you decide to take it apart there are springs under pressure inside of it. I wouldn't advise doing it unless you need to change out some part of it because of failure. Bolt goes back in the same way it came out.
The action and barrel can be removed from the stock by removal of probably 2 screws on the underside of the stock, I am sure some require more. Once removed the action can be pulled away from the stock. The action will have the trigger group attached to it. The savage accu-trigger is adjusted by a special little wrench they provide, other adjustable triggers are probably adjusted in a similar manner. Don't take the trigger assembly or safety apart unless you know what your doing, but simply taking the action off the stock will give you a pretty good idea of whats going on. The action will still function with the stock off, cycle the bolt and pressing the trigger a few times is worth it to know how it operates.
The most basic answer for why bolt actions are more accurate than semi auto rifles is that there are less moving parts, expanding gasses are only being used to drive the bullet in a bolt action, but a semi-auto uses expanding gas to shoot the bullet and drive the action.
Don't be afraid to take the stock off the action, it will require only an appropriately sized screwdriver. It goes back on the same way it went on. In the best case scenario, these screws should be torqued down to a specific tightness. I generally go until they are not going to go anywhere, but without forcing them. Even snugness for both of them.