To make a medium and large battery you will need a small battery, the question is how to make it?
You don’t. Both the grid scale and personal device scale small batteries are only found as loot, or from disassembling the appropriate devices. Both rechargeable and non-rechargeable small batteries are beyond the abilities of a survivor to realistically craft, we simply lack the tools and raw materials.
The grid scale small storage battery is pretty rare, but you’ll usually find plenty of medium and large storage batteries in electric vehicles, so its rarely an issue. As for the batteries for personal devices, depends what version of the game you are playing. If you’re playing an experimental version from before August 2nd or so, or if you’re playing the 0.H beta, you can reliably get the high capacity version from smartphones which are extremely abundant.
If you are playing a newer experimental version, there is an overhaul of batteries underway. Battery capacities are being brought down to sane energy density along with devices having reduced power draws, but its still a work in progress. Light rechargeable cells appear to only be a lootable item now, as smartphones now disassemble into disposable batteries - this is likely just an oversight as the new standard is worked through.