How do I know when to use lead, flux-core, lead-free, or any other kind of solder out there? Do you have any tips on solder gauge for specific applications?
A great question, and since a textbook could probably be written to answer it, there's probably not going to be any single answer. I want to provide a general answer tailored to hobbyists, and hope that people more knowledgeable can come in and tie up specifics.
In general, if you got it at the electronics hobby shop, it's good to use for hobbyist purposes.
Lead vs Lead-free
60/40 lead solder melts at 315C and takes about 1.5 seconds to melt and form a bond, aka "wet". Good bonds are shiny and shaped like a "tent", not a ball. With just a little practice, you can get good or at least very competent using lead solder.
However, lead is a toxic heavy metal, so extended skin contact isn't good for you (and terrible for the environment if disposed of improperly). I like to wear very thin cotton gloves while soldering (but I don't always). Note that lead does not "evaporate" during soldering. The smoke you see is the flux. But you shouldn't be breathing flux either. I use a fan and filter when soldering. There is also the "breath out" technique to avoid the fumes, which is fine for small jobs.
Lead-free solder melts at 340 to 370C (depending on the formula), and takes about 4 seconds to wet. Good bonds are not shiny, and it is harder to visually detect a bad joint, at least at first.
Simple answer: unless you are planning to sell the device you are soldering to someone in the EU, stick with lead-based solder. Lower soldering temperature and faster wetting time of lead-based solder means less chance to thermally damage your board and parts (and its cheaper). Electrically, you can use either. You can even use lead solder to rework a lead-free board. Of course, then it won't be RoHS.
Diameter of solder
Very thin solder, 0.020" dia or less, gives you a lot of control over how much solder you are putting down, and melts a bit faster. But you have to "feed" the solder into the joint at the right speed, and unrolling another foot from the roll every couple joints gets old. I sometimes don't get enough solder into the joint with thin solder because I don't/can't feed it fast enough. Good for hand-soldering fine SMT parts.
Thick solder, 0.050" dia or more, is good for making big joints, like heavy gauge wire or leads on a TO-220 regulator. But it is easy to put down too much solder, and seems to melt slower as the solder itself acts as a heat sink.
I generally prefer "mid sized" solder, 0.025 - 0.031" dia, for most work. It gives me a balance of controlling how much I put on a joint, without the hassle of feeding the hair-thin stuff.
Electronics flux can be rosin, water-based, or no-clean. All are about the same quality as far as de-oxidizing the copper so a good solder bond can be made.
Rosin flux leaves an ugly, sticky residue. Cleaning it off takes either a LOT of water, or a (nasty) chemical solvent. You shouldn't leave it on, as it is slightly corrosive, and can also be somewhat conductive. This is falling out of usage due to the environmental impact of cleaning.
Water-based (aka Resin) flux is less ugly, not sticky. The stuff I have used leaves a white film behind. I have heard this film can cause problems with long term reliability. Some people just leave it on, but removal takes only a moderate amount of water.
"No-clean" flux is resin flux that burns or boils off, leaving almost no residue.
"60/40 lead solder melts at 315C"
Rubbish. 60/40 Sn/Pb solder has a pasty range of a few degrees C and solidifies at 183 deg C. A eutectic solder such as 63/37 Sn/Pb melts and solidifies at 183 deg C.
"However, lead is a toxic heavy metal, so extended skin contact isn't good for you"
Again, not true. Lead cannot propagate through the skin. It can only be ingested by transference from fingers to food or cigarettes, or through inhalation if atomised.
"Lead-free solder melts at 340 to 370C"
An alloy such as 96.5Sn/3.0Ag/0.5Cu is liquidous a t 217 deg C.
"Electronics flux can be rosin, water-based, or no-clean. All are about the same quality as far as de-oxidizing the copper so a good solder bond can be made."
Wrong. Different fluxes have different levels of activity and will help wetting and coalescence to different degrees.
"No-clean" flux is resin flux that burns or boils off, leaving almost no residue."
These are commonly synthetic fluxes that do not "burn off' but remain after the joint has solidified and are inert so do not pose any long-term risk to the circuit from hygroscopic action or ionic contamination.
It's a tricky topic to get good information on, but lead-free solder's supposed to be worse for you, overall.
Yet Wikipedia claims
You also have to worry about tin whiskers