Assuming your OEM struts are OK i would start with anti-sway bars. they will tighten the body roll and increase maximum cornering speed, After that i might look into a stiffer spring set with new struts. Strut bars are inexpensive but they do more for the looks under the hood than performance, especially if everything else is stock.
on a 12 year old car your stock struts are probably pretty worn out so I would start by checking them first, If they need replacing i would do a performance strut rather than a coilover kit. your choice of OEM height springs will be a bit limited but there are a few performance companies that sell springs to match their struts, for OEM ride height with some performance adjust ability KYB makes some really cool struts.
13 WRX Hatch - Torqued Performance Stage 1+
Slow is Smooth > Smooth is Fast