This is my first moba, and I jumped into ranked as soon as I hit lvl 30. I placed in silver 2, fell to silver 5 pretty quick, and hovered between silver 5 and silver 4 for a few months. Eventually dropped to bronze 2.
I got out of bronze maybe 3-4 weeks after that, and I ended at gold 5 that season. The next season I placed at bronze 2 and climbed out in about a week. Ended at plat 4 that season. This season I placed in silver 2 and have hovered around there for a few months, but I'll be back in plat by the end of today or tomorrow.
My advice is to pick 2-3 champs that can cover 3-4 roles. Those champs should not have extremely high pick/ban rates. My picks in season 5 were Teemo for top/jg/adc (this is no longer advisable) and Zyra for mid/supp. If you focus on expertise with a few champs, you'll very quickly outskill players in bronze and low silver.
I picked my mains based on a variety of factors. I wanted something I could play most of the time, so that meant champs with low pick/ban. I wanted something that could play well from behind, so that meant champs with some kind of fallback utility. I wanted something that wasn't too reliant on teammates, so that meant champs with adequate 1v1 damage. And lastly, I wanted something that could perform decently against both tanky and squishy teams, so that meant champs with a mix of burst and dps.
Champs that fall into those categories today might include:

Most of those champs can fill multiple roles, have a decent variety of build options, are low pick/ban, OR have a decent combination of burst and dps. Most of them are not currently meta, but most also tend to exist sort of outside the metay.
Like Fiddlesticks hasn't been meta for years, but his dedicated mains can crush in jungle, top, mid, or support. Kayle can play literally any role in the game, but she was only meta in jungle for a couple months since I started League. Zyra hasn't been meta in a solo lane since season 4, but I still mained her in mid and top for 3 seasons.
If you follow that advice, you will climb faster. League changes a lot, and a big part of climbing is how fast you can adapt to change. It's easier to adapt to changes on a few champs than a lot of champs, and it's easier to improve your game on a champ you already understand. Good luck!