I have an old transistor radio with three or four wires disconnected on the circuit board and headphone jack. I know the radio works because I held the wires onto the terminals and listened for a connection. Anyway, a couple of the connections are on the circuit board, the other two are on the headphone jack. Would it be difficult to just solder them on myself? Are there any particular precautions I need to take? I have a very old "hotshot" soldering gun, the tip got hot, but not enough to melt the solder on the board. Thanks

Depending on the age of the radio you might have to sand the places where your going to solder if it has a coating on it (sometimes shelak) or wax is on it. You can pick up a soldering iron for fairly cheap at most electronic stores. I think even Wallyworld sells them too if you don't have a RadioShack nearby.
Also make sure you use rosin core solder and not acid core. Acid core is usually for binding metals. The other (rosin) is used for electronics.

You say "not hot enough to melt solder on the board",
After cleaning the joint, touch solder to the tip, to verify it will melt the solder, then wipe off with a damp paper towel/sponge.
The you heat the joint and apply the solder, too hot and the trace will lift off the board. Not hot enough, you will get a cold solder joint, usually dull, yor connection should be shiney. Maybe do a search on Youtube or Metacafe on soldering. Good Luck.

Also the leftover solder seems to be conductive... if this means anything; do I need to get some new "flux" or whatever its' called?

Solder IS conductive, If you have a desoldering iron, remove the old solder, then solder with new:
http://www.radioshack.com/product/index.jsp?productId=2062731
You can also use a Soldapult:
http://www.google.com/products/catalog?q=soldapullt&rls=com.microsoft:en-us&oe=UTF-8&startIndex=&startPage=1&um=1&ie=UTF-8&tbm=shop&cid=14334315704258392658&sa=X&ei=nLXXTbe2Ioe-0AHn36n8Aw&ved=0CDoQ8gIwBA&biw=952&bih=629#
Or braded copper wick, you could use copper shielding from a piece of coax.

Ok thanks