Weapon and armor racks don't do anything right now because they were buggy, eventually they will be fixed to hold weapons and armor.
If soldiers have access to the weapons and armor you assign, they should run and grab them. Sometimes it takes a while because they are busy doing something else, but if it's been more then a month or so there might be a problem. Did you accidentally lock a door between them and the weapons or something? If not, check the items and see if they have TSK next to them, that means there is a task that wants to use that item - for weapons or armor it would probably be hauling it to a stockpile. They won't pick it up to use it if a task wants it, so you will have to wait till someone gets around to putting it in the stockpile first.
The only reason not to give your dwarves the best armor available is because plate mail is kind of heavy. Armor user skill lets a dwarf ignore part of the weight of his armor, and strength lets him carry more before he gets slowed down. I usually just suck it up and give recruits steel plate anyway but alot of people like to give them something lighter to train them in first - it's up to you.
As for weapons, like others have said there is a risk of the dwarves getting maimed if you give them nice weapons right at the start. When sparring, there will always be accidents sometimes - highly skilled dwarves (or very tough ones) can either block the blows or dodge them, or if they get hurt recover from them. Raw recruits, however, will likely get their spines broken or arms lopped off or something equally terrible if you just set them all up with masterworked steel weapons. Instead, either give them a shield but no weapon or give them weak weapons to train with till they get good shield and wrestling skill.
Material matters, yes. Silver does 50% of the damage of iron, copper does 66%, bronze does 75%, Iron does 100% (of course), steel does 133% and adamantine does 500%(!). Armor has the same values. Also, the quality of the weapon or armor effects it too - 'no modifier' is 100%, masterwork is 200%, and the others are in between. So you might want to give your dwarves no modifier silver weapons to train with (just don't forget to give them real weapons when the goblins come!).
The exception to this is crossbows. The quality modifier works when they fire bolts, but the material does not matter - a masterwork wood crossbow works just as well as a masterwork adamantine crossbow. When dwarves run out of ammo, however, they will usually just charge the enemy and beat them on the heads with the crossbow, and material DOES effect the damage done from that so giving them steel crossbows is not a bad idea.