The very first thing that PS Vita needs is a price drop. From this point on, Sony should always have an under-$200 option on the table, even if it's just the core system. However, I'd recommend they package a system with a memory card (at least 8 GB) and a month of PlayStation Plus, and sell for $199.99 or less. People appeared to be quite receptive to the Black Friday deals. Sony needs to work on making great packages like that the norm, and not the exception for just one or two days a year.
The hardware design is mostly fine. I'd add HDMI out, and make the Select and Start buttons easier to press (slightly raised, instead of flush with the surface).
As far as developer support goes, they need to court more iOS and Android developers, and convince them to port their hit games to PS Vita (for high-end games) or PS Mobile (for low-end games). If publishers feel it's too risky, or developers don't want to take time to learn a new system, then Sony should consider licensing the rights to a few of the games themselves, and assigning a hand-picked developer (either internal or a "trusted" third party) to handle the conversion. By doing this, Sony gets the benefit of having the big-name games, while the original developer/publisher can relax with minimal risk and let the money roll in.
Sony also needs to grow PS Mobile in general. I believe that PS Mobile could be Sony's true long-term "portable PlayStation" platform, but the pieces aren't all in place yet. Vita is the short-term solution, and it's a necessary step toward the long-term solution of PS Mobile taking over. There are several things they need to do to fix PS Mobile as well, particularly granting certification for more devices from more manufacturers, but that might be another topic for another thread.