How do you choose soldering iron tip size and solder wire diameter?

#2 chisel tip is the best for most circumstances.
the real fine point ones rarely work as you might expect. The tip heats to a point, but that point isnt the utmost tip. For soldering ram or real fine work, maybe, but most often Ill find the hot spot and trim the tip to the hot spot. for soldering tssop/ soic chips get the solder paste. tack the  corners with the chisel tip.

for long chips, pin headers, something with a long line of pins, tack in the first and last pins, and let the structure of the thing work to your advantage, holding the thing in place, as you work back. if not, you might find a tendency for things to shift in ways impermeable until its a pain to fix later.

through hole stuff is pretty simple. leaded solder works quite well, ROHS solder stinks, none of this stuff is particularly good for you. dont eat or drink stuff while you solder, and wash your hands  after.

I go with 0.031in flux core ROHS solder typically. I run the iron a 850 degrees Fahrenheit. which is arguably kinda hot, but when you get better at soldering, running a hotter iron will let you move faster.

I typically deposit a small bead of solder on the tip, hit the pad, and use the tip of the solder to push the liquid metal around the pad.

There are as many ways to solder as there are folks soldering, so just figure out what works for you.

Get a fan to pull fumes out of your face.
wear safety glasses. You get this liquid core solder, the center heats up fast, and may have a tendency to pop beads of molten metal out in unpredictable places.

if you get sharp icicle looking bits of solder forming when you pull the tip away, it means your iron isn't hot enough relative to your solder.

The multi tip kits are pointless, except to find the tip you are most comfortable with using. its cheaper to just find what you like, and get a few of those.

get yourself a scrub, and tip tinnier. starting out, I'd be amazed if you didn't destroy a few tips. leaving them on, dry, for long periods, exposure to corrosive fluxes, oxidation.... youll figure all that out, and how to avoid it as part of the normal learning process.

A temperature-controlled soldering iron helps, mine is a 40-year-old Weller TCP-1 with the PU-1D base. I mostly use two bits, a number 7 chisel-tip and a number 7 pencil-tip. Sometimes I use a long pencil-tip which is currently buried under bits and pieces for a project I am working on so I can't read the number just now. The base has an added on/off switch and power-on indicator light that doubles as a 'temperature correct' light. I may have to upgrade both iron and base when I run out of tin/lead solder.

Removing components from old boards is good practise, you might need a solder-sucker though as I find the copper-braid solder-removal ribbon is not very good. Be careful with the sucker, they can pull the track off the board if you are unlucky. If you can test the components after removal and find that they still work then you are doing well! Although chips like Z80's and 6502's will be difficult to test off-board!

I always use rosin-cored solder, even when I'm got my plumber's hat on and a blow-torch in my hand. It's OK up to 15mm copper tubing.

Patience is a virtue, you'll get bad joints if the iron is not up to temperature. From the photo's it looks as if you are improving all the time (assuming they are shown in old-to-new order of course). Good luck!

A temperature-controlled soldering iron helps, mine is a 40-year-old Weller TCP-1 with the PU-1D base. I mostly use two bits, a number 7 chisel-tip and a number 7 pencil-tip. Sometimes I use a long pencil-tip which is currently buried under bits and pieces for a project I am working on so I can't read the number just now. The base has an added on/off switch and power-on indicator light that doubles as a 'temperature correct' light. I may have to upgrade both iron and base when I run out of tin/lead solder.

Removing components from old boards is good practise, you might need a solder-sucker though as I find the copper-braid solder-removal ribbon is not very good. Be careful with the sucker, they can pull the track off the board if you are unlucky. If you can test the components after removal and find that they still work then you are doing well! Although chips like Z80's and 6502's will be difficult to test off-board!

I always use rosin-cored solder, even when I'm got my plumber's hat on and a blow-torch in my hand. It's OK up to 15mm copper tubing.

Patience is a virtue, you'll get bad joints if the iron is not up to temperature. From the photo's it looks as if you are improving all the time (assuming they are shown in old-to-new order of course). Good luck!

Yes, I did the double row connectors first, then the single row connectors.