Play through at your own pace. Never feel like you need to compete with anyone.
If your playing on bnet, keep any gems you find, and merge them up to pgems using the Horadric cube (3 gems of the same type and grade = 1 grade higher, eg 3 chipped rubies = 1 flawed ruby). You can use perfect gems to trade for some pretty good gear.
If you play through the game for the very first time, concentrate on the story and try to play through it without any help, except you are stuck somewhere. Don't be afraid to make the wrong decisions when assigning skill points etc. Your very first character isn't meant to be a killer in the highest difficulty level anyway. Most advice, no matter how well-meant they are, will spoil much of the fun of the game. It was the most fun for me when I was a newbie and didn't know a thing about it.