Sure thing. In the meanwhile, this is the Cliff Notes version:
1. if you follow the cable from the shifter towards the hub, it probably has a housing stop clamped to the frame, and then the bare cable continues on and passes over a clamped-on pulley. Make sure both of those clamps are snug so they aren't sliding down the frame tube, or it'll keep going out of adjustment.
2. where the cable meets the hub, there's a tubular adjuster that threads onto a little threaded rod. The rod pulls a tiny chain that disappears into the end of the axle (on the variant I'm describing, that is). If you put the shifter into first gear, then use your finger to gently tug on the cable, the tiny chain should be able to move just a little bit, like 1 millimeter, in and out of the axle. If not, thread the tubular adjuster up and down the metal rod until you've got that slight movement in first gear.
One other prerequisite, and a caution: if the hub itself moves, this changes the effective adjustment on the cable, so you also want to ensure your axle nuts are snug. BUT DO NOT OVERTIGHTEN THEM, particularly the left one, because they're prone to stripping and they're hard to find replacements for.