It's all to do with the settings on your machine

You need to have it set to zig-zag stitch, and when it's set to that, the stitch length dial will set how far apart your stitches are (so you need to set it as low as possible - every machine is different, there's a sweet spot between so short that it starts to lump up, and just too long that gaps show between stitches.) And the stitch width setting (which also usually determines where your needle sits i.e. left, middle or right) needs to be at the widest. It's this that you change if you want to make your band of applique narrower e.g. when you're coming up to an edge.
All the usual things with setting thread tension apply as well, if the tension is too tight then it'll bunch up and go all lumpy, if too loose you'll have loose threads that get caught in the feed dog.
The only other advice I can give you is that when you're turning a corner, pivot the fabric when the needle is on the same side as the direction you're pivoting. What I mean by this is if you're coming up to a corner, and you have to swivel the fabric to your right to present the new edge to sew down, make sure the needle is down on the right. This makes a nice neat corner, whereas if you do it with the needle on the left there'll be a gap. Same applies for curves too!
Other than that, it's just practice. Applique is horrendously monotonous but space out for a few moments and you've got a lump or you've run out of bobbin thread or some other dumb thing. So it's quite tiring to do for long periods. I would suggest a number of practice shapes first till you get the hang of it. I figure you're not wanting to rush this costume anyway, not with that level of detail

You will need a metric ton of thread, I went through four or five regular Gutermann cotton reels making Ibis, and you've got far more to do than that if you decide to go for it! Make sure you bondaweb your shapes down prior to sewing too, makes it so much less hassle!
Here's a sample tutorial but you'll want your stitch length to be much shorter than hers so there's no obvious zig-zag going on:
http://sewlikemymom.com/applique-tutorial/