I do not like bore snakes .... after their first use, it's like washing your hands in dirty water. You can actually put more damage causing residue in your bore than you remove.
Here's my suggestion .... buy a good one piece cleaning rod with a bore guide. The proper diameter rod will be 13/64" for a 22 cal bore and will be threaded for 8x32 accessories such as bore brushes, patch holders, mops, or jags. I like solid brass cleaning rods, in fact I make my own on a mini-lathe but most any metal or carbon fiber one piece rod will do just fine. I avoid 3-piece aluminum or steel cleaning rods because they can come unthreaded when used and the open joint can be damaging to the bore or crown. Bore guides look like a cone with a hole drilled in the center. The concept being .... you hold the bore guide tip in the bore and by keeping the cleaning rod centered, it will not touch the crown when used properly. So much for the cleaning rod.
I always start off with a liberal amount of powder solvent .... brand is not important but I use Hoppie's and have been for more than 50 years. Other brands should work equally well as long as it says "Powder Solvent" on the container. I actually pour some solvent in the bore then use a wet "bore mop" to distribute the solvent throughout the bore. This step is called "mopping" .... you can also use a clean cloth patch that fits loose in the bore to apply solvent. To work properly, the bore must be totally wet from end to end. After mopping, let the gun sit for 15 minutes. This will allow the sovent to dissolve powder residue and will help loosen lead fouling. Some people use CLP products (clean, lubricate, and prevent rust), however these products don't do the individual tasks near as well as powder solvent and gun oil.
The next step is to use a bronze bore brush, again wetted with powder solvent. Lock the bolt back so the bore brush can exit the chamber. Never try to reverse the direction of the bore brush while it is still in the bore. Take about 5 full length back and forth strokes from the muzzle and make sure you use the bore guide. This should loosen all powder residue and most lead fouling. All 22 LR ammo uses lead bullets .... some bullets are thinly plated with brass or copper making them more foul resistant, however even plated bullets can result in lead fouling. Lead fouling is almost impossible to see unless it is really built up. What happens is a bore brush makes lead fouling shiny and it blends in with the steel bore. It is a good idea to check a bore a few days after cleaning because lead will corrode quickly and turn a dull gray, making it much easier to see. If you do discover more lead fouling, repeat the above process. If there is still lead fouling, mop the bore with penetrating oil and let it sit over night. Penetrating oil will actually creep under lead fouling and make it very easy to remove. Kroil is an excellent product for removing lead fouling in this manner.
After brushing the bore, use a patch holder and a clean white cloth patch .... old "T" shirts (jersy material) works great. Push the patch all the way through the bore and back making sure it fits snug. Remove the dirty patch and use a fresh patch to get the rest of the fouling out with another full stroke. Last, apply a few drops of gun oil on another fresh cloth patch and do one more full stroke. This is excellent for rust prevention.
All 22 semi-autos get dirty inside the action. This is because 22 LRs are "blow back" operated, meaning the built up pressure in the barrel will push the spent case and the bolt back so the bolt can strip a fresh round from the magazine and chamber it. During bolt movement, a lot of powder residue crud is blown back into the bowels of the gun and spread to every nook and cranny. Eventually, this crud will build up to a point where the gun starts to malfunction. You don't have to clean the internal parts every time you clean the bore unless you want to but I recommend doing a full field strip every brick of ammo (500 rounds) and doing a thorough cleaning. For a 10/22. the best way I have found is to remove the two receiver pins, then remove the trigger group. Removing the recoil spring and bolt is a bit more difficult so if you aren't too mechanical, you can access most of the areas that need cleaned while the bolt is still in the receiver. Flood the bolt, receiver, and trigger group with powder solvent then let them sit for a few minutes. Scrub the breech face and any areas you can see with a tooth brush and more solvent. Use compressed air to blow out the dissolved powder residue and powder solvent. Every couple bricks of ammo, you need to remove the recoil spring and bolt so you can clean the powder residue from the firing pin channel and other areas that were not accessible.
Once the internal parts have been cleaned, place a small dot of oil on each accessible part and use a clean patch to spread the oil to form a very thin film that feels dry to the touch. Using too much oil is about the worst thing you can do with a 10/22 or any other gun because oil will actually draw powder residue and form a very abrasive sludge. Oil has the opposite effect of what you might think. It turns gummy with age and rather than reduce friction, it actually increases friction, much like molasses. Using a very minimal amount of oil will prevent rust but it will not draw powder residue nor will it turn gummy. Excess oil is your worst enemy .... especially if you shoot in cold climates.
Many people are conditioned to believe using large quantities of oil is a good thing. I can assure you this is just the opposite, in fact when I owned my gunsmith shop, about half the guns I got in for repairs were nothing more than a result of excess oil. In most cases, a good cleaning put them back in service but there were occasions where parts were actually worn out due to the abrasive sludge and had to be replaced. I can't stress using a minimal amount of gun oil enough!! Also, I never recommend using automotive or household products for guns. Many oils have added detergents that can eat bluing and leave small pits in the metal. Granted, ounce for ounce, gun oil is way more expensive than automotive oil but if you follow these guidelines, a 2 1/4 oz bottle is about 4 bucks and should last for a long time. I clean a lot of guns per year and a 4 buck bottle of Hoppie's gun oil will last me 2 or 3 years.
Here's a photo of some 1 piece brass cleaning rods with bore guides. These happen to be for handguns but a rifle rod is exactly the same only longer and don't photograph well. The top rod is for a 17 cal, the next is for 22 cal, and the 1/4" diameter rod is for centerfire guns. The short rods are for cylinders. The longer rods have "free spin" handles: