yes. look at your weapon, it will have a number, say it does 50 damage, and then your skill level will have real numbers, like (20-25) next to it.
Running, jumping and such use less stamina, by a LOT, when you are skilled up.
Those are the 2 main effects, weapon damage and stamina use. You can look at the wiki for any other details, but I would not even attempt some bosses (eg, bonemass) without proper skill (here, blunt weapons in the 40s or so).

Arrows... there are 2 or 3 playstyles here. A casual archer may lob a wood arrow at a deer or use it to anger a distant enemy to fight on better terrain. A more serious archer would get a sneak attack hit to do 3x damage before swapping to melee, weakening the toughest enemy in a group or whatever. And of course the full time archer.
For anything more than potshots at deer or the like, wood arrows are useless: use something else cheap to make that does lots more damage. Fire is cheap to make and useful for quite some time. After that, I just move to needle and the newer mistland ones which are also plentiful and easy to make. Your limiting factor here is feathers, but birds are all over, and a harpoon takes a bird ammo free, or a wood arrow works here too. In that gap between fire and needle, if not playing a full archer, you can FIND enough arrows in the iron mines to keep you covered usually, or make just 1 stack of something intermediate.

Anyway, you can make do with vastly inferior equipment, depending on how you want to play, but good luck one-shotting a fuling with a wood arrow.