I'm not the best barbarian but I've experience what you have and am keen to help you with this. I used to have pure strength gear and dual-wielded till I got owned by packs of white mobs in normal mode. Here's a breakdown of what you should do to survive. Dual-wield VS Sword and Board
Do note that dtrength gives a dmg% modifier to our outgoing damage and also provides armor. Dualwielding 2 non strength weapons may may result in less dmg than a socketed weapon with a strength shield.Follower
Use a Templar with this build for all modes and give all your hand-me-downs to himhttp://us.battle.net/d3/en/calculator/follower#0100Normal Mode
For a decent build, you can try thishttp://us.battle.net/d3/en/calculator/barbarian#bcYViT!aYX!ZZZaaa
Use Cleave for trash mobs and Bash for bosses as your primary skill. Switch your Rend/Seismic Slam to Hammer of the Ancients for secondary skills.
For gear, use str + vitality gear which you can get from drops or AH. Since leveling is easy, you can change gear at every 10-15 levels.
I recommend getting used to and either dual wield or sword and board(my preference).Nightmare Mode
A slightly modified build as compare to Normal Mode
Gear stat priority remains the same but updated every 5 levels or so.
Sword and board is highly recommended for this. Also, obtain weapons with sockets and use red gems to increase damage output. Hell Mode
I'm currently progressing thru Hell Mode and I find that some blur or yellow elites will steamroll you with ease rather than it being the other way round. Hence I'll recommend doing some self trial and error with Elective Mode. From here modify and play around with things. See what suits you best.
I hope you have a better understanding and guide to life as a Barbarian in Diablo 3. Knowing all your barbarian's skills and passives is crucial to playing the game. Feel free to add my battle tag and pm me ingame. I'm still learning myself so what I put may not appeal to others. Hope it helps you though.