Here's how to field strip (take apart) a Ruger 10/22 rifle for cleaning.
Here are the few tools you'll need:
Small thin screwdriver or a punch
Step 1: Safety First
Take off all accessories (scopes, sights, etc). Remove the magazine, open the bolt, and ensure there are no rounds in the chamber.
Step 2: Safety button and screws
Push your safety so that there is an equal amount showing on both sides. If you don't do this, the button will get stuck on the stock as you go to pull it off. Next, loosen the screw in the center, but don't take it out. Then loosen the screw on the barrel band and take it off. Now lift off the stock, and all you have left is the barrel/receiver assembly.
Step 3: Remove pins
Depending on your rifle, the pins may just fall out on their own, or they may need some help. If they don't come out easily, gently use the screwdriver or punch and a hammer and gently tap the pins out. Put them somewhere where they won't get lost. Don't let the pins come out of the trigger assembly, unless you want to. I show how to disassemble the trigger assembly here: www.instructables.com/id/How-To-DisReassemble-the-Ruger-1022-Trigger-asse/
Step 4: Remove trigger assembly
Take the trigger assembly out and put it aside.
Step 5: Removing the bolt and spring
Pull the bolt handle back and lift up on the front of the bolt, and it should just come right out. Remove the spring as well.
Step 6: Done!
This is as far as you need to go to clean the rifle. If you want to, you can remove the barrel by using a hex key to remove the V-block that holds the barrel in place.
Step 7: Putting it back together
Put the bolt spring back in. Using your hand and a screwdriver, pull the bolt handle back all the way. Take the bolt and push it into the spring, you should hear a click.
Step 8: Attach trigger assembly
Put the trigger assembly so the holes line up and insert the pins. If the pins don't want to go in, gently tap them with a hammer.