Required Tools
Soldering Iron
A soldering iron and some solder are the main tools you'll need. We discuss specific recommendations in the free How To Solder book. The super short version is: buy an inexpensive iron from a marketplace (Amazon, Ali, Ebay, etc). It'll get you through a handful of pedals before you decide whether it's worth upgrading. The quality won't make a difference if you're using good techniques.
Flush Cutters
Or diagonal cutters, or side cutters, or wire cutters.
These are for snipping the excess leads after soldering resistors, capacitors, and diodes. You can find these individually for a few dollars, or bundled in kits.
Sponge
A damp (not soaked!) cellulose kitchen sponge will work great for keeping your soldering iron tip clean. To determine if it's suitable, make sure it's the type that gets hard when it dries. That's a pretty good indication it's cellulose. Bath sponges that are always soft, however, aren't going to work and will melt.
Another popular option is a brass soldering sponge. These are made of tiny strips of brass that don't thermally shock your iron tip the way a damp sponge does, so it doesn't experience as much temperature drop. Usually in practice it's not a big deal, so don't worry about it yet. Plus the brass sponges tend to leave little balls of solder on your tip which can be annoying.
And finally, a folded pile of damp paper towels can work great in a pinch too!