There's really no need to be nervous about cleaning.

Generally, you should always clean a firearm from the breech. Theory is that running a metallic cleaning rod through the bore from the muzzle may inadvertently damage or prematurely wear the rifling at the crown, imparting uneven forces on the bullet as it leaves the barrel, thus degrading accuracy.

Some designs are obviously harder to clean from the breech than others, lever-actions, slide-actions, and semi-autos in particular. Though you probably CAN remove your Marlin's bolt, you may also consider using a .30-caliber "Bore Snake" instead. If your .22's bolt isn't easily removed, you may want to purchase a second one in .22-caliber. Being flexible and nonmetallic, save for the brass (softer than steel) weight and the bronze brush embedded, these feed through the breech easily. They're not the perfect solution, but they're a fairly low-cost option if you're not mechanically-inclined.

In reality though, my first centerfire was cleaned from the muzzle for a couple years until I figured out how to remove the bolt. I religiously cleaned it after every use with a common metallic sectional cleaning rod. Accidental damage and premature crown wear aside, it still shoots 1 MOA or better at the ranges I typically hunt with it at.

Use a good solvent (Hoppes is almost synonymous), apply a light coat of lubricant to prevent rust, keep solvent and lube off any wood, glass, or painted surfaces, and you should be good to go. When you've dumped thousands of dollars into a precision rifle, or shoot hundreds of jacketed rounds each year, you can begin to worry about things like chamber guides, one-piece rods, copper fouling, alloy jags and brushes, etc. For now, just get out and shoot.