Now, you may (or may not) be thinking...I'm okay with straight bias tape, but now I've come to a corner! Oh god, what do I do!?
Well, do not fear, because there's an easy way to do them cleanly.
This is what I usually see, and what you don't want to do. This is my first ever sewing project (Kamui jacket), and as you can see, at the time, I didn't know what to do either. So I just mowed over it with my sewing machine and hoped for the best. Unfortunately, this approach usually results in a "best" that is significantly underwhelming.
So, let's try this out instead. I'd recommend doing a trial run first with some scrap fabric and a pen your first time through. Once you're used to it, you can go without.
Take your bias tape, line it up, and mark where the seam will be on your fabric corner. Do the same from the other direction, and you will have two lines guiding you for the next step.
Take your bias tape and stitch along the fold line.
Stop EXACTLY at the intersection you just drew. You may want to decrease your stitch size as you get close. Precision is KEY!
For the next step, you'll want to take the side of bias that's not stitched down and fold it at a 90 degree angle at the corner. This should align the rest of your bias tape with the other edge of the fabric.
Stitch until the intersection, just like you did before.
It should look like this from the bottom.
Keeping your fabric folded the way it was when you did your second line of stitching, draw a horizontal line across the fabric that matches up with the first line of stitching. After you do this, draw two diagonal lines up and down to form a triangle.
Fold the rest of the fabric except for your bias tape away from your triangle.
Stitch the corner fold in, using the lines of your triangle as a guideline.
Check the fit if you like, then trim the excess fabric away so it can lie flat.
Fold your bias tape over the corner. You might want to use a point turner of some kind to make sure that the top is nice and crisp. It should look like this on the front.
And like this on the back!
At this point, you need to do something to secure the back. You can:
a) stitch it down normally
b) stitch in the ditch
c) slipstitch it (prettiest method, but most time consuming)
d) slipstitch it then topstitch it
e) fabric glue/stitch witchery
The choice is yours!
Now your bias tape looks great from both sides! If you slipstitch it, it will look even nicer. (see last two photos)