Say I have Boartusk Liege and Dralnu's Crusade in play. Boartusk Liege gives all other red creatures +1/+1 and Dralnu's Crusade says "all Goblins are black and are Zombies in addition to their other creature types". I assume that means that Boartusk Liege does not give +1/+1 to Goblins?
You are correct: when both cards are in play, Boartusk Liege does not give Goblins +1/+1. A ruling on Dralnu's Crusade says
The Goblins are both Goblins and Zombies and they are black (they lose their previous color(s)).
Since they lose their previous colors, they are not red.
In regards to how this works in general, rule 105.3 says
Effects may change an object's color or give a color to a colorless object. If an effect gives an object a new color, the new color replaces all previous colors the object had (unless the effect said the object became that color "in addition" to its other colors). Effects may also make a colored object become colorless.
The reason that the Goblin doesn't get +1/+1 from Boartusk Liege before you change its color to black is that Magic uses a layer system to determine in what order effects get applied. Effects that change color are on layer 5 and effects that change power and toughness are on layer 7, so the color changing effects get applied first.