I’ve already posted about the actual tools, but in order to solder, you also need solder. In general, I recommend the thinnest flux-core solder you can find. Most of the time flux-core solder removes the need to use separate flux, which simplifies the process a lot. The thinner the solder is, the better your control of the amount of solder will be. I’m not really an expert on this, but here is what I use:
I’ve been using this for a year or so now. Before that I was using leaded solder. Leaded solder will always work the best, but lead is an insidious and poisonous metal. In addition, it’s not legal to sell electronics with lead in it in a lot of places anymore. This stuff solders almost as well as leaded solder. It’s a bit on the pricy side, but it last a really long time.
If you use large spools of solder, like the one above, it’s helpful to have a spool holder of some sort. It works well enough that I never have to think about it.
Sometimes, you do need flux, and if you do, I recommend liquid flux over flux paste for most things. Liquid flux allows for smaller thinner applications, which I think gives better results. It’s also easier to clean off afterwards. Flux is corrosive, so you shouldn’t leave it on your electronics once you’re done soldering.
If you’re doing surface-mounted soldering, then you need solder paste. For many simpler designs I don’t use stencils anymore, I just apply solder paste directly from a syringe onto the board, place the components and heat it up with hot air, then I measure and fix anything that doesn’t look right. I’m not sure which solder paste is best, I have a couple of them and this is one of the ones I use. Unlike regular solder, solder paste tends to go bad after a year or two, so if you do this sort of things, you’ll end up buying new ones every now and then even if you haven’t used up what you have yet.