Is my journey to soldering just going to be a lot less painful if I go and get an FX951 or something?
Yes absolutely. A very long time ago I struggled with electronics soldering (I was good at pipes and radiator, even using irons) until I got an Antex 50W temp-controlled iron (now sold as the TCS50W, but unfortunately for me only in 230V trim) and then suddenly and magically, I could solder.
So, IMO, the minimum for doing decent soldering is a basic temp-controlled iron or station
that has good quality tips. Antex (irons) and the Hakko FX888D are examples--these are the least I would consider but there are other good options in the $100 (US) bracket. Next step up would be the basic T12/T15 cartridge-style irons, like the FX951, KSGER, etc. Then you can spend more and more, but those upgrades will be incremental compared to that first leap from absolute crap to a real tool.
Your little tub of gelled flux is fine, but I'd also get some brand-name eutectic solder. I can easily manage either way, but for delicate or mechanically tricky operations eutectic is so much nicer to work with that I don't mind paying $$$ for a spool.