One of the keys to easy soldering is having a good soldering iron.
You want one that holds the proper temperature. That means it should have a thermostat and lots of power. The use of larger tips (more thermal mass) also helps when working on things like the TG terminal lugs...
I use a 
Weller
[WTP series
iron. Not cheap though --- a good iron will probably cost $70-$150...
You can solder pretty much everything (small things, big things) with a good soldering iron, but it does help to match the tip size to the size of the work, so get two or three different sized tips for your iron.
The other option for things like solder on the TG solder lugs (but less useful for more delicate work, like working inside amps...) is a good soldering gun (lots of power, no thermostat, but you turn it on with a trigger and rely on timing to prevent overheating.)
When desoldering wires on your Hammond, you'll probably want to use
a _solder sucker_ to remove excess solder.
The Edsyn "SoldaPullt"
is the canonical brand.
"Solder wick" is sometimes very useful for cleaning solder off of more delicate parts. It's generally useful for working on circuit boards, and other things which are daintier than that found in your typically tonewheel Hammond.
Weller has a nice little pamphlet on soldering:
Here's another guide to soldering:
And here's a whole collection of learn-to-solder articles and videos:
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