:wikipedia:
erm, correct, but the most important aspects of weapon skill benefits aren't there :V
The most important things in melee are
base damage (which gets multipliers as you skill up) and
attack speed (which also goes up with skill and lets you apply the multiplied base damage more often in the same amounts of time).
Every two points in a weapon skill reduce delay by 10%, down to half the base delay or 70% for two-handed weapons (some short blades have different conditions but I don't remember the details since short blades are good for stabbing and not much else anyway). This means you want to get your weapon's skill up to 1/10 of its delay as soon as possible, so you get the minimum delay with it. This also means that if you find an executioner's axe on D:1 it's still advisable to wait a little to use it since at extremely low skills you either OHKO everything or miss and take multiple hits while "recovering". The damage multipliers are a mystery even to some devs, though; general consensus is that after you max your attack speed, weapon skill gives diminishing benefits, even though said benefits are not negligible.
Unarmed combat works differently; you start with decent speed and low damage, and as you raise the skill it becomes increasingly ridiculous up to the point where at max skill your fists are probably the fastest AND the most powerful weapon you'll ever find.
Fighting helps a little with all kinds of melee but the main benefit is the HP gain; either way there's no harm done in letting it train all the time since apparently the "where will this amount of exp go" check for weapon skill first and fighting later, which means it'll often be lagging behind anyway.
And on spellcasting skill, it's important to remember
how the spell hunger reduction calculations work (the wiki is a risky place to hunt for tips since there's a LOT of outdated analysis and a decent amount of bullshit there, but it's very reliable when it comes to raw data). Increasing spellcasting also increases spell level memorization slots.
I didn't proofread any this and I edited it a lot on the go, so if something just plain don't make any sense, ignore it :V