THE best way to win, in Bronze and Silver Leagues is to have one GOOD deck and a few good neutral cards to add to the free cards to make other decks, when you need to.
The best way to do that? Rent cards. Rent all the cards you need for one deck, which is all you will play until you get to at least Bronze II, when they start limiting the Splinters you can use.
The BEST decks- for Bronze, Silver and even Gold Leagues, are a little pricey- even to rent. Those decks are:
Yodin- Yodin is again, a pricey card- even to rent but the good new is, the rest of the deck is pretty reasonable. You put a Tank up front (Cerebus/Living Lava are good and F2P) then use Ranged monsters with Snipe/Sneak to hit the soft center of their team. The L3 Fire Beetle and Ferexia General are great additions to the F2P cards because they are ranged attackers with 2 Attack and Snipe. Another good attacker to add to your deck: Pyromaniac. He doesn't have Snipe but he does have Sneak- which lets him target the BACK spot in your opponent's line-up.
Other cards to consider adding: Flamesmith, Ettin, and Grenadier. Their rent is CHEAP- the Flamesmith and Ettin make great back spot characters while the Grenadier is another 2 Attack (ranged) monster for 3 mana. Again, in Bronze, Silver and even into Gold Leagues- this deck CAN'T LOSE, once you get above 27 mana.
Llama/Kron- This deck is UNSTOPPABLE! at 17-25 mana. Just play it and win- it's that easy. Unfortunately, the prices and rents on those 2 cards are sky high so it may not be economical until Silver, at least. Still, if you add in a L3 Flesh Golem and just use the F2P cards- you should breeze through every game from Bronze to Gold (as long as you can use the Earth Splinter and have at least 13 mana) you, literally, CAN'T LOSE until your opponent gets ~30 mana and has a good deck.
Some other, good, and less expensive options for cards/decks are:
Mylor Crowling- FANTASTIC, inexpensive and fun. A L1 Mylor with the F2P cards can win a lot of games in Bronze and Silver. Once you get to Gold, you'll want a L3 to go with a L3 Flesh Golem- until then, a L1 is great. Some cards to consider to go along with him would be: Nectar Queen- she's 9 mana but she got high health, high attack and flying so she makes a great Tank, Sand Worm, Unicorn Mustang, Wood Nymph, Javelin Thrower, Beatrix, Flesh Golem and the Headhunter- those will give you some attack/healing to go along with your tank.
Fire/Melee Deck- very common and very powerful in Bronze/Siler leagues, the summoner is F2P (Malric) then you just load up your team with a FAT tank and melee monsters who can attack from the back lines (Reach, Opportunity and Sneak Abilities). Some good cards to get to use in this deck are: Battering Ram, Sand Worm, Serpentine Spy, Giant Roc, Phineas Rage, Goblin Mech, and the Chicken.
Water- the Water deck is cheaper, more flexible and not as common in Bronze and Silver. It's more flexible because you can play it different ways- Magic Value or Melee.
Magic Value option is: Turtle, cheap magic attackers with Alric as your summoner. The Sea Genie, Coral Wraith, and Dwarven Wizard are all great and inexpensive cards to add to this deck and while the Mischievous Mermaid and Magi Sphinx are also great, they run a bit more. A great upgrade at Tank, or even in the 2nd position in the deck is: Lobstradomus. For 6 mana he has 9 Health + Shield, this will keep him alive against everything you'll see in Bronze or Silver while having a Magic Attack means that Alric will boost him too! The final cards for the deck would be Spirit Miner and The Phantom of the Abyss- pricey but well worth it as last position protectors.
Water/Melee Deck features The Demented shark plus an all melee team- Wave Runner, Electric Eels, Battering Ram, Feasting Seaweed, Sabre Shark, Elven Cutthroat, Sand Worm. Very effective and with leveling some cards, great right up to Gold.