Just to add to all the good advice already...
I would really recommend a soldering station as it makes soldering a lot easier on a variety of different surfaces. It can sense when more heat is needed and adjust it to keep the tip hot and you can adjust it to solder everything from terminal lugs to a chassis to surface mount components.
Here is the type we use and it has worked great on everything from tube type equipment to very fine circuit board repair.
http://www.cs-sales.net/sl30.html
Replacing bad caps and wiring takes care of most problems especially if the items have some functionality already. Yes you do want to replace these old components because they will continue to degrade and will eventually fail, especially if the device is left on for a long period.