Where is the Hue selection?

strange to have logo from photoshop, i suggest to redo it on illustrator, logo must be a vector file not bitmap.

In order to use Recolor artwork: is the object selected?

It's a great feature, you should definitely see if you can get it to work. When applying it to black objects, you need to set up its recoloring options so that black is allowed to be recolored.

Apart from that: with the plugin Phantasm you could also have Hue/Saturation in Illustrator (the plugin is not for free)

Is it a vector object or an image that you are trying to color? Images are better  adjusted in Photoshop.

Is the logo vector paths, or is it a raster image? If the latter, look for a source for a vector version or rebuild from scratch. Or edit the color in Photoshop.

Peter

In Illustrator you would select an object and click on a swatch to apply that swatch color to whatever is active at the moment, either fill or stroke. There are other ways of applying color, but that's kind of the easiest ways if you're starting.

If the swatches panel is not visible, bring it up by clicking on Window->Swatches Menu