Ehhhhh ... for me, general rule of thumb is:
3,000 for dino oil
5,000 for synthetic blend
10,000 for full synthetic.
And the filter is the cheapest part and one of the most critical ones. I don't see the idea behind dumping $30 in oil and sparing a $5 oil filter, other than laziness.
Also, never use anything less than what is recommended for your engine, meaning, if full synthetic is recommended, ONLY use full synthetic.
If synthetic blend is recommended, you can use that or full synthetic, but don't use dino oil.
Personally, I always use synthetic.
If you are towing or racing, change it more often (but since this is a Corolla, probably safe sticking with what I posted above).
In my '08 Matrix, I did full synthetic changes every 5,000 miles anyway, and there was 0 sludge in the engine after 75,000 miles of hard driving. You can't change it too much ... but you CAN change it too little, and damage the engine severely.