I've been thinking about pulling the stock 350 out of the 'Velle and building her into a 388. (350 block, .060 overbore + 400 crank) I was just wondering if this is safe or will this engine be prone to overheating? Would it be better to just build it into a 383 instead?

Also I'd like to extract about 550 HP from this engine....what all would be in order to do this?

1. I would do the 383 and bore it to the minimum requirements to clean up the cylinder walls. That way you will have some material for the next rebuild. Their is not to much of a difference in a 383 and 388. You will not feel the extra .030 of power and you can consider that .030 as a deposit into a saving account for the next rebuild.
2. Good budget with some good heads, carb, intake, and cam. But make sure your suspension can transfer the power to the ground.

When a .030 bore is done on any sbc do you really not feel the difference? I always thought you'd feel something....of course then again I guess 5 extra cubes don't really add up to much?

The .030 overbore is to bring the cylinder back to a perfect round shape/diameter and not as a performance addition. Adding a longer stroke will increase hp/tq more than a .030 bore.

Does the same apply for .040 and .060 overbores?

I was aware of boring an engine to get the cylinders truly round again but wasn't aware that it had little effect on performance.

You are correct, want more cubic inches adding stroke will increase it more than .040 or .060 over bore.