Required tools  

For assembling most of the kits all you need is common hand tools and a soldering iron. For testing an analogue or digital multimeter comes in very handy...

Soldering
The soldering iron should (of course) be a type meant for electronics use. For soldering the kits with SMD ICs you will need a fine/narrow tip. Many of the kits include a single or a few SMD resistors and/or caps. They are large types that can easily be soldered with a standard tip. If a kit includes SMD ICs, it's mentioned on the kit page.

Solder
Use standard 60/40 solder (60% tin) or use "silver solder" (62/36/2 or similar) - but don't use lead-free solder. Lead free solder is harder to use, requires higher soldering temperatures and each solder-joint takes longer to make. This extra heat can easily damage some of the parts supplied with the kits. I choose parts because of quality - not heat tolerance :)

Test equipment
As mentioned above, an analogue or digital multimeter is usually sufficient. But access to an oscilloscope is always nice for trouble shooting.

The only exception is currently the FM MPX decoder kit. For aligning the filters on that board you need (access to) a signal generator (sine waves only) and a VTVM or 'scope.