Early Melee Weapons:
The absolute best place to quickly get some great, early (I am still using it tens of hours in), melee weapons is with the Champions in Whiterun. After a few easy quests you get your pick of Skyforge Weapons given to you.
Early Armour (and OK Bow):
A very good early armour set (for both light and heavy armour users) can be acquired by joining the Imperial Army in Solitude (it is quite a trek from your starting location, and you do have to take out a fort of bandits by yourself first).
Also the Imperial Bow is a good starting bow and you should be able to pick one up off of a dead comrade in one of the earlier missions.
Also this quest line yields a lot of easy missions that sees you fighting along-side Imperial troops and fighting armies of Stormcloaks.
I am not sure if this is guaranteed or just random loot but the best bow I have ever seen, a Supple Ancient Nord Bow, I got off a mid mission enemy encountered in an early Mage College mission. One downside is that it currently cannot be upgraded any further, so while it is one of the best bows available, many bows (with some good smiting and materials) can be made a whole lot better.