The Fuzzy Select tool only selects areas of similar colors. To select the entire star, after clicking the black outline hold down the Shift key and click the yellow interior and each of the eyes.
Or, perhaps easier, first use Rectangle Select to select a rectangle around the star, then switch to Fuzzy Select and subtract the white area around the star from the selection by clicking it while holding down the Ctrl key.
Once you have the star selected, I'd suggest moving it to a separate layer so you won't have to repeat the process the next time you want to move it. Perhaps the easiest way to do that is to cut the star out with Ctrl+C, paste it back with Ctrl+V and click the New Layer icon in the Layers window to turn the resulting "floating selection" into a new layer. (Or you can use the Select→Float menu command (Shift+Ctrl+L) to float the selection without cutting and pasting, then turn it into a new layer as above.)
Do the same for the text, and then use the Align Tool to align the two layers as Jack suggested.