Starting Hand
As a general rule when playing vs. any class, always assume it's the most aggressive version of that class. Slower versions give you more time to draw deeper into your deck anyway, so it's better to be wrong vs. a control deck than an aggro deck. You should keep Fiery War Axe and Justicar vs. any opponent because of this. Aside from this, you typically want to prioritize whirlwind effects and brawls vs. decks like zoo/paladin that swarm you with lots of minions. Single target removal (bash/shield slam/execute) is better vs. decks like shaman, dragon warrior, and druid that play one or two large
Once you're in Legend, the player pool is small enough that you can keep tech cards vs. decks you know your opponent to be playing. That said, for general ladder climbing, I believe the above strategy to be correct.
Aggro Shaman
The goal is to kill everything ASAP and preserve your life total. It's great to get down to 11-12 life to turn on your revenges, but make sure you've got Justicar or a Shield Block handy to armor up once you've done so. Aggro can burst you down from as high as 21 damage with enough mana and time to draw into the combo, assuming they haven't used any rockbiters. You want to be using your bashes, axes, and slams on the everything with 3-4 toughness and saving the executes for their 7/7s or minions that they tried to hide behind taunts.
Don't be afraid to use two cards to kill any 4-toughness or divine shielded minions, as they're suiciding card advantage vs. you every turn. Your hero power is card advantage vs. them. Eventually you'll run them out of threats and they'll start stocking up rockbiters, lava bursts, and hoping to burn you down in one turn... which will usually fail since you're gaining 2-4 life a turn and hitting them for 8-10 by then.
Dragon Warrior
This matchup is very favorable. You have answers for everything that they play, the most they can burst for is 15 from an empty board (and that's only the ones that run inner rage), and they're slow enough that you don't fall behind too fast before you're able to stabilize. Their only real out is getting Life Tap from an early Finley, and even then it's only slightly unfavorable as shield blocks and acolytes allow you to draw about as often as they hero power. You spend all game trading 1-for-1 with their minions, until they eventually run out of relevant threats. Don't be afraid to be greedy with your answers. Don't waste executes or shield slams on 3-toughness minions unless you have no brawls and are going to die, and don't worry if you're only able to brawl away 1 or 2 creatures. That's standard as Dragon Warrior isn't really a deck that "goes wide". It's nice to save brawls for when they drop a larger minion, but as a rule I typically brawl whenever I'm taking more than 6 damage a turn. 