Why did my first solder joints on speaker wires come out so rough?

The dull gray spot is almost certainly a cold solder joint, which means you didn't get enough heat into the connection itself. The lumpy look you are seeing is likely due to adding too much solder at once rather than melting a proper bond. Don't worry, this is a common learning curve when moving from just touching wires to actually forming a reliable electrical connection. For the mechanical side, you are right about needing a holding device. Trying to hold the wire steady with your fingers while simultaneously trying to manipulate the iron is nearly impossible. A cheap clamp or even a hobby vise helps keep the component steady while you work. Also, you need to strip your wires much shorter than you probably did. Twisting the fine copper strands togather after stripping them helps consolidate the material before the soldering process, which makes the joint much stronger and less prone to slipping. When dealing with the thin wires coming off the speaker cone, those are often delicate and may have thier own terminals that need attention. If they look like small, bare wires emerging from the cone material, you should definitely treat them the same way you treated the main speaker terminals. Use flux to clean and prepare the area, strip the leads, and ensure you heat the wire and the component pad at the same time before applying the solder. Only apply the solder to the joint itself, not to the iron tip.

If the connections feel solid when you tug them, you likely succeeded electrically even if the aesthetics are imperfect. A clean joint is about the metal fusing, not about how shiny or neat it is. Just focus on getting the heat and the solder interaction right, and you will see massive improvements.

My big worry was probably that the iron wasn't hot enough to get the metal to stick together properly. I kept watching the wire and making sure it didn't look like it was getting burned, so I guess I was scared to get it really hot. Does the part of the iron that touches the metal need to be shiny or is that part of it not important for the connection to actually happen? I feel like I was just guessing how much heat was needed for everything.

I think worrying so much about wheather the iron tip needs to be shiny is overkill. You can get a joint done fine even if the tip looks a little crusty if you follow the proper technique. The real failure point isn't the iron itself, it's the lack of heat transfer to the actual metal where you are joining things. If you just sit the iron on the wire and pad, you might melt the solder but you haven't heated the component enough to make a good connection. You need to heat the contact points first.

I found that my biggest issue wasn't necessarily the amount of heat, but rather the surface area of the iron tip that was making contact with the component pad itself, since just touching the wire wasn't enough to bring the entire joint up to the requisite temperature. Specifically, I always tried to touch the iron to both the terminal lug and the wire simultaneously, effectively creating a bridge of heat transfer before introducing the solder to that specific area. If you are using an older iron with a very small tip, you really need to make sure that the tip is actually clean and that the solder is melting onto the joint material rather than being absorbed by the tip of the iron itself.

I remember when I first started, I was really worried about burning the pad if I left the iron on too long. What actually worked for me was just getting the wire and the pad hot together for a solid couple of seconds before the solder even touched them. Once the metal was hot enough, the solder just seemed to flow and grab the surface perfectly on its own. I always try to clean up any excess before moving the iron to the next spot.