You should be able to solder most joints in bass/effects applications, with the possible exception of grounding on the back of pots with a properly prepared 15W iron. I have used a 20W orange cheapy Weller for many many things and it works well for the $15 it costs. I now have a nice temp controlled station, but here are some tips to get it done on the cheap.
1. Clean joints - use some solder wick to clean everything up if you are reworking old parts. The surfaces must be as clean as possible to make a good connection. New parts and old parts can be cleaned with isopropyl alcohol to remove oil and contaminants.
2. Clean tip - yes, make sure your tip is clean before you attempt the joint. Make sure the tip has been properly tinned and cleaned before every joint on a damp sponge. If you don't know how to tin the tip or what this means, google it. Easy.
3. Flux can be your friend - solder flux (not acid plumbing flux but rosin flux) etches the surfaces about to be joined and makes the joint flow much better. Not necessary for all joints but helps immensely in larger joints (i.e. large ground connections) and may help you "cheat" your way to a good joint. I now use a dropper bottle but have used and still have a flux pen I got for a few bucks. Both work.
Make sure your iron is hot, ready to go, and all surfaces are clean. The cheap pencil irons can work well but the tips wear out much quicker and they don't hold as much heat. Most important - practice. Find something that sucks and disassemble the board, use it to practice installing components, etc.