How do I resize a part without the part going on top of the object it is inside of?

You would use CFrame for that purpose; it's an absolute position (I guess..?), while with Position, it calculates if there's an object, then set the position to be above the object. (CFrame stands for Coordinate Frame.) Here's the WIKI document in case I didn't explain clear enough. (I recommending looking it up anyway, as my information may be incorrect.)

Link: http://wiki.roblox.com/index.php?title=CFrame

I assume you're using a loop to make the sphere become larger in size. Once you've familiarized yourself with CFrame, using the link TheeDeathCaster provided, you can then simply set the sphere's CFrame to remain constant as the size increases with each loop.