You can find better weapons and armor in shops, but they will start appearing sooner in the hands of the bad guys. Bandits or Marauder dungeons are your best choices when you are looking for new gear. Bandits wear light armor, and Marauders wear heavy. They all use the same assortment of weapons though.
If you are playing a warrior type character, enchanting your weapons and armor is a necessity as you reach higher levels. The only way to do this in the vanilla game is to join the Mages Guild and complete the Arcane University recommendation quests. Once you have access to the Arcane U, you can use their spellmaking and enchantment altars. Or you can get the Frostcrag Spire dlc, which also has the same, and does not require you to join the guild. Once at an altar, you put your item on it, then a filled soul gem. This is the complicated part. You can buy empty or filled soul gems from merchants, usually the Mages Guild alchemists in every city has a few for sale. If you have an empty one, you have to cast a soul trap spell on a victim, and then kill them while it is in effect. Their soul is then svcked into the gem. The stronger the soul, the stronger your enchantment will be. Note that soul gems come in varying grades, and you need one at least as high as the level of the soul you are trying to trap.
Because of this many people do the Azura Daedric quest, as the reward is a reusable soul gem (the normal ones are destroyed after you use them to enchant with). Most people also usually add a 1 second soul trap effect on their weapon enchantments, at the end of the other effects they pick.