So you need:
rosin core solder, NOT acid core. usually it will say on the package for electrical use, and its generally of a smaller diameter than the solder used for plumbing apps.
soldering gun- 80watts is good, anything higher is just faster, but some have duty cycles (limits to how long they can be on before needing to cool down) the pencil types are nice, as opposed to the gun types
wire stippers, the ones in the pic are better, the ones with the crimp features are OK but tend to be of lesser quality. When they are closed, look to see how tight the gaps are, and try and move around the actual cutting parts to see if they have play/will bend.
electrical tape or liquid electrical tape, heat shrink tubing is nice as well
ends, wire, etc
The colors of the eyes and butt connectors reference the size of wire that goes with that connector, look at the packages at the store and match them to your wire size.
Its good to have/leave some solder on the end of the torch,
I remove the plastic shit off the ends, it tends to come off anyhow, and you'll need it off if you solder this joint. A twisting action on the eye while pulling is the key,
Matching the wire gauge to the correct hole size on the strippers,
Squeeze through the insulation, but when you go to pull it off, I pull farther back on the insulation, as to not drag the cutters along the copper, which tends to remove some of the strands with the insulation.
Notice a nice snug fit, but probably a bit too snug to get the metal wings to clamp on it real good. People will have differing opinions on this, but this terminal was probably at the max of its wire gauge range here,
Twisting the strands, then inserting them into the connector, notice how I stripped too much, so I cut some of the excess copper strands off,
I went up to the next sized connector (the yellow ended one) for the other end, similar prep, and then clamped them both down. Notice the overlap, I think that is stronger, imo..
Heating the ends of the strands, not the connector, let them heat soak,
Then dabble on the solder, start with getting some on the gun and feeding it to the wire. Once its going on the wire, don't add any more to the gun, but walk it back as the wire heat soaks, letting it melt in through the folded-over metal wings of the connector, and then to the back by the insulation,
The other end with larger connector,
Note I kept the solder mostly off of the eye where the screw or bolt will stick through, as thats not really necessary,
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Now to do a direct wire splice, which is preferable to using connectors as there is greater strength and less resistance, start by stripping the wires back. Note here, this is thicker wire, so I ended up stripping more back. Normally about 5/8 to 1/2 is good enough. Cross them about in the center,
And fold them around each other at the same time, evenly. I don't have three hands, so this is what your end result should be. Notice how the ends of the wire pretty much wrap up and stop just as the insulation begins, and both stop at about the same place at the joint, if everything is cherry,
Resting the soldering gun tip with the angled part on the joint, not just the end butted up to it, for better heat transfer, I let it heat up for awhile, then begin to feed some solder into the area between the torch tip (the hottest part) and the wire joint. I get a bit on the tip of the torch and kinda walk it down to the wire, but then I just add it to the wire from then on. If it doesn't flow well at this point, it needs more heat. Don't add too much solder to the tip of the torch, as it won't heat transfer well then. If too much solder gets on it, remove it and a quick flick of the wrist and the solder will come flying off onto the ground, your bench, your friend, whatever.
Feed the solder to the points of the ends of the insulation, don't go overboard but cover up the copper nicely, walking the solder around on the wire so it spreads evenly. The heat will do most of the work, as well as the flux thats built into the solder. Finished product,
After it cools, then I tend to put on liquid electrical tape, usually two thin coats will do it well, and then if its a spliced joint, regular electrical tape over it. Reason being, sometimes if you have a sharp point of wire, the liquid tape will not cover it well, so a single layer of regular tape ensures sealing.
People also use heat shrink, which is great, but if your wires are short, the stuff tends to shrink up on the wire because of the soldering heat before you can slide it over the finished product, then heat it up. Thats a matter of preference, as long as its protected well, then you're good.