I have a 10/22. It's a fine carbine.
It's better to learn how to use iron sights first, and then move up to a scope. With a .22, you aren't going to be shooting at targets 500 yards away, so it's ok to go with a cheap scope. The factory sights are probably "bore-sighted", so they should be reasonably close. They may need some fine tuning. Also, note that every different kind of ammo will shoot to a slightly different spot.
Barrel length has surprisingly little to do with inherent accuracy. More than anything, barrel length determines velocity and sight radius. A short, stiff barrel can be more accurate than a long thin whippy barrel. A longer sight radius also makes a rifle with iron sights easier to shoot accurately. Barrel lengths vary quite a bit, but for .22's, they tend to be on the shorter side.
Theoretically, a synthetic stock can be better. A synthetic stock is basically weatherproof and more scratch resistant, while humidity over time can potentially cause some wood stocks to warp. However, a cheaply made synthetic stock can be floppy and cause accuracy issues, and it's also possible to waterproof a wooden stock. Personally, I don't think it really matters and is more of an aesthetic issue than anything.
You may also wish to consider an air rifle. You can get a lot more practice in if you can just go shoot in the garage or backyard whenever you want.
Do you have any other particular questions?