Need some more details. What field(s) are you looking to go into? Are you moving more for opportunities or for lifestyle? Either way, definitely have a job offer in hand before moving, unless you have a lot of savings or family money to keep you afloat while you look for one.
Austin is a super hot destination for people in their 20s (and beyond) and as a result it's become quite expensive. So budget is a huge factor here. Assuming you are a typical recent grad, don't try to live downtown as it's mostly luxury high rises, and even the older apartments are exorbitantly expensive.
If you want to be in a cool part of town that is in or near where there's a lot going on, I'd look at East Austin (though it's getting more pricey every year), South Lamar, the triangle. The further you get from downtown, the cheaper it is, unless it's west. There are some great new apartments off of Manor/MLK. South of 71 is also more budget friendly than north of it.
