Do I really need Desoldering for replacing resistors/caps?

You can use old solder, but fresh one is always better.
If you spot cold solder joints or cracked with magnifying glass, than you simply can re-touch the old one .
try to go http://www.parts-express.com/cat/desoldering-tools/520 and choose 45W desoldering iron.

I use Weller, but it’s hard to find. Had purchased mine 10 years ago 4lb.
I would not recommend neither tin or ones with silver added. Make sure you're bellow 200-Celcius range of melting Will be impossible to work and will require high heat which you often don’t want to use on circuit boards.

There are some places where RadioShack is still opened. I was still wondering if the place 15 miles away from my home in north of Durham NC is still the only one active RadioShack in the world!
They sold similar model to this one and that's the one I use professionally. 
http://http//www.parts-express.com/ecg-j-045-ds-45-watt-desoldering-iron--372-202
Desoldering pump is important because of an idea of removing old solder and replace it by new. Have flux ready as well to make sure that solder fills up and clean the surfaces before soldering back.

I used $5 solder sucker and figured that desoldering iron works substantially faster and better.
Using solder sucker might require extra helping hands.