I noticed an issue with my gears switching recently so I decided to try indexing them. I've gotten to the point where going from the smallest cog to the largest works perfectly, but in the other direction it doesn't want to move from gear 1 to 2. If I keep shifting up it shifts normally, only a gear behind, until it gets to gear 8 and the chain skips the 7th cog. Does this point to an issue with the derailleur itself?
This sounds like classic derailleur cable friction. The cable gets stuck inside the housing and stops or delays the derailleur moving to smaller sprockets. This is caused by dirt contaminating the housing, or perhaps corrosion of the cable.
The fix is to remove, clean, re-lube and re-install the cable, or replace the cable and housing altogether.
Welcome to Bicycle Stack Exchange. As Argenti Apparatus suggests, the most common cause of ghost shifting (which is the issue of switching gears you refer to at the beginning of your question), and problems shifting--especially down the cassette, low to high--in general, is excess cable friction. Lots of causes such as dirt within the cable housing, corrosion or fraying of the internal cable, kink or other anomaly (flat spot) of internal cable especially in the areas where they enter or exit housing, ferrules (metal, plastic or rubber end-caps on the ends of the lengths of external housing) can also wear and start to hamper the internal cable's movement, and if your bike directs the cables along the bike's downtube and underneath the bottom bracket shell, the bottom bracket cable guide can be a source of excess friction. Another possible cause of difficult shifting to higher gears (larger to smaller sprockets) can be excess cable tension. In this case, turning the barrel adjuster clockwise a half-turn at a time and checking the shifting between adjustments (basically its an aspect of indexing, i would say), may be the answer. Obviously the excess friction should be ruled out first. Links to a few references: Servicing/tensioning a rear mech https://www.parktool.com/blog/repair-help/rear-derailleur-adjustment https://www.sheldonbrown.com/derailer-adjustment.html