"Coilover" is short for "coil spring over shock". It consists of a shock absorber with a coil spring encircling it. The shock absorber and spring are assembled as a unit prior to installation, and are replaced as a unit when the shock absorber has leaked. This provides for optimal damping without torsional loads. Some coilovers allow adjustment of the ride height and preload, using a simple threaded spring perch system similar to a nut. More advanced adjustable coilover systems will use a threaded shock body, coupled with an adjustable lower mount for height or "lowness" adjustment, while an adjustment knob is used to adjust the damping or "stiffness."
Springs alone are cheap. Doing a spring and strut combo properly costs about the same as a good coil over setup. Expect to spend in the $600-1000 range for decent coil overs. The $1000-1500 range usually offers a lot of options.
If you were to get springs, once the stock struts have blown from the increased wear of riding so low you will need coil overs to replace everything. So sort of yes, if you get springs, you will need coil overs, unless you want to get the HFP suspension kit or a custom Koni based setup.
You have 2014 Si same like me so there aren't any coilovers available eyet. Last night I ordered Eibach springs, but once good quality coilovers are out then I will get the coilovers.