So I usually play in Gold / Diamond, but I also rent a lot of cards and wanted to understand what cards were most valuable to the lower tier players, so I tried to work my way through bronze (and perhaps silver next) using the minimal number and cost of rental cards possible. I finally did it after a couple days of playing and testing and I'm here to share my results.
The image above shows off most of the lineups that worked best to rank up (just ignore that one loss in there). In order of what I thought was most effective:
Quix The Devious + Agor Longtail + Healing:
This worked surprisingly well in many battles where there was enough mana for one healing monster and one armor repair monster. (Minimum mana cap is going to be 25 with Quix) Most lineups just lack the sufficient firepower to take out a healing Agor at bronze level. There is an example in my screenshot where Agor lost, but in that case, the enemy team was able to output 15 damage per round, something that is fairly difficult to do, but the enemy lineup was also relatively fast. In other matchups (like the top battle on my list) there is a mix of fast and slow monsters. Because of the flying ability, many attacks by slow monsters tend to miss, giving the healers a chance to catch up. Overall, I won many more battles than I lost using this lineup, and really this only requires the Quix summoner and Agor monster. For healing, I often used Scavo Hireling, Venari Crystalsmith, Merdaali Guardian, or Adelade Brightwing.
(Ilthain / Owster Rotwell) + (Thorns/Reflect/Return Fire Ruleset):
Whenever a ruleset came up for thorns, reflect, or return fire, using Ilthain or Owster made the battle much more predictable for me. Choosing the right summoner with one of these rulesets made it such that the enemy really could only use one type of attack against me effectively, and if they guessed wrong, they take a lot of extra damage. The second and third battles in my list demonstrate this.
Aquatus + (Djinn Oshannus / Phantom of the Abyss):
This lineup is somewhat random because of backfire, but I'd say it definitely won me more than it lost. I'd prefer Oshannus over Abyss where the enemy may use magic because of the void ability, but if the enemy is forced to use melee, Phantom of the Abyss seems to do better because of his higher speed and dodge ability. Also goes well with a heal, but enemy sneak ability really counters that heal. Regardless, sneak monsters will eventually have a tough time against Oshannus or Abyss.
So these three lineups and a couple of tips I'm about to share pretty much gets you past bronze.
Rent some cheap, high mana cost monsters to fill a lineup. Costs will vary, but the exact card doesn't matter so much in my opinion, just whatever looks powerful and cheap. Dragon splinter or neutral work best for this so you can use them with the most splinters. These cards are for the 99 mana cap rulesets. At bronze, I feel I am mostly playing against bots, and even in the max mana rulesets, they just keep playing "regular" lineups using the starter cards. If you fill up your lineup with large monsters, even if they're not particular synergistic, that alone pretty much guarantees victory.
At bronze, I feel like there are limited strategies that are employed by the bots. One of their favorites is to have a lot of sneak monsters. My guess is that new players tend to not look out for attacks at both ends and are still grasping the idea of having a tank in front first. This makes sneak very effective against newbies. There are a few ways to counter this. Usually sneak monsters don't do that much damage, so using a monster at the back of your lineup with shield or thorns sometimes works well. Another strategy is to copy the sneak strategy and to go heavy on sneak monsters when you feel you have an advantage somehow based on rulesets. Monsters that are used very often for this strategy are: Uraeus, Silent Sha-Vi, Tenyll Striker, Serpentine Spy, and Stitch Leech.
Have a "go-to" cheap tank for each splinter.
Fire = Antoid Platoon or Living Lava.
Water = Cruel Sethropod or Diemonshark (but often times Djinn Oshannus alone with Aquatus is good enough).
Earth = Hill Giant or Unicorn Mustang. (I actually didn't use Earth that much in bronze).
Life = Pleacor Conjurer or Blinding Reflector.
Death = Crypt Beetle or Cursed Windeku.Other cards I sprinkled in and won me a round every once in a while: Raa, Carnage Titan, Runemancer Kye, Corsair Bosun, Chanseus The Great, Oshuur Constantia, and Coeurl Lurker.
You don't need to buy cards for every splinter. I concentrated on dragon, life, water, and death. I pretty much ignored fire and earth. Dragon was definitely used the most. I used death and life next because of Owster and Ilthain, and I pretty much only used water for Oshannus + Backfire combination.
I hope that helps some newbies trying to crawl out of bronze. I know it can be frustrating, but the bots have a weakness. They rarely use the newer cards because there is less data. They rarely use the entire mana cap because there is less data. And finally, they tend to play the obvious "optimal" lineup for a given ruleset and can be predicted and countered fairly often.
Here's also some stable diffusion images of Splinterlands characters I tried to make, but they kind of failed, but I still think they look cool and wanted to post them somewhere. :)