Always wear safety goggles!
We solder all the time at work as we work on medical equipment and replace components. Really you don't need expensive stuff to solder correctly. If I didn't have any soldering equipment, I wouldn't spend more than $30-$40 for everything I need. Here's a short list:
1. Soldering iron (be it a hand held or a station....as long as it reaches 600-700°) You can use a metal lid/bowl to set your iron in
2. Flux - I like the paste tube because you will have better heat flow throughout the components giving a quicker/even flow of the solder around everything. Sure it's not the funnest stuff to clean up, but a few wipes with alcohol on a rag and your done.
Nothing wrong with rosin-core solder, I just prefer the paste
3. Solder - I honestly haven't seen a difference in any solder I've ever used. Maybe I've not used the cheap stuff....dunno. lol I use what I've got, which I didn't pay much for.
4. Solder wick - it's a weaved copper mesh. You use this to remove the solder from a component so you can remove it to install another. Anyway, put the wick on the solder joint and put the iron to it.....the wick will "suck" out most if not all of the solder. no flux needed using wick.
Here's a few things to take note of.
When soldering wire together, I put flux paste on all exposed wire. If the heat from the wire stops taking in the solder (your flux is basically gone), stop and add some more.
A clean soldering iron is your friend. If it has a filthy tip, it's not going to transfer the heat as well. It takes 10 seconds to clean and tin your iron. Let it get up to temp, lightly brush it with a metal brush if it's super filthy. If it isn't rub the iron across a wet rag/sponge.
After that tin your tip. You do this by applying solder to the tip. Then take your iron to the wet rag/sponge again wiping off any excess solder. It will look nice and smooth/shiny after this step. That's how you know you've got a clean iron that isn't gonna waste your time trying to heat up the component/wire.
If your soldering something small, (one handed method of soldering) put a little blob of solder on your tip and add it that way. You need more, do it again. Larger stuff (two handed) you'll need to get it up to temp and feed the solder into it with your other hand.
lol look for some crap electronics and play for a bit, you'll feel like a pro in a few minutes. There's technique, but you'll figure that out quickly.
Ton's of video's on Youtube.
Hope this helps