Your instructions should have paint references, but otherwise Tamiya's TS-3 Dark Yellow, XF-61 Dark Green and XF-64 Dark Brown seem to be popular choices. The TS-3 is a spray can though, think it's XF-60 for the "jar" version. Personally, I use Vallejo Model Air colors, they do a 1943 Panzer Dark Yellow, a 1943 Panzer Dark Green and a Tank Brown. Then there's Model Master, Mr Gunze, and so on... It's down to personal choice and what you can easily get hold of.
The Vallejo paints you only have to mix with their own thinner, same with the Tamiya ones.
As for an airbrush... Everybody has their choice. I use an Iwata Eclipse, but then there's Badger and Aztek (and probably others). As long as you get a fairly decent one you should be ok. I use a 0.3mm nib which is good for acrylics. Then you need a compressor, preferrably with a tank, and so on. You'll find literally hundreds of threads in here on airbrushes and compressors though so have a look around. If you've never airbrushed before it's a good idea to get a cheaper model than your 3-in-1 to practice on first though.
Good luck with it all.