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I haven't been on a bike in years, and just picked up a used one to get into riding (2013 Specialized Hardrock).
The rear gears aren't shifting properly, it seems to only move between the 2 smallest, and I noticed that when the shifter shows it in 1st gear, the chain is actually on the smallest cog. Am I right that something is wrong here? And what's the easiest fix for this?
Almost certainly the gear shift cable has slipped through the pinch bolt that holds it onto the derailleur, so there is too much slack in the cable, and the shifter cannot pull the derailleur onto the larger (lower gear) sprockets. This is pretty easy to fix:
Note which side of the pinch bolt on the derailleur the cable goes. Undo the cable pinch bolt and remove from the derailleur.
Make sure the cable and housing are seated correctly in the shifter and frame stops.
Check cable and shifter by shifting gears while keeping tension on the cable - the shifter should pull and release the cable as you shift.
Shift to highest gear number, pull cable taught, screw the barrel adjuster on the derailleur all the way in, then back out a turn, do the same for the adjuster on the gear shifter. Attach cable back to the derailleur. Now the shifter should be able to pull the derailleur onto the largest sprockets.
Now you can adjust the limits and indexing so the derailleur places the chain onto each sprocket properly. It's hard to describe how to do that briefly in an answer here so check see this Park Tool page which explains how it works and walks you through the process.
