@Daekwan Since you're in DC and therefore need tires that can handle freezing weather in the winter, here's my advice:
First, you won't find a tire that can hold up to warm dry track use (once you get any experience / speed) and freezing weather on the street. Doesn't exist.
You
could do 3 sets of wheels+tires of course, 1 for track, 1 for summer, 1 for winter. If you end up liking track days and want to do this, go for it, obviously it gets you the best of each world. But also obviously, it's extra money up front (3rd set of wheels), and you need to store
two extra sets of wheels at a time, and you need to swap wheels for each track day (though honestly that's trivial with other track prep you'll probably be doing, e.g. pad swapping, frequent brake fluid changes, checking torque on everything, etc).
Now let's say you prefer to just have 2 sets of wheels+tires. That's very doable, and that's what I had back when I was doing track days + driving in the snow in the winter, with one car.
One option - which is what I did - is have a summer street+track set, and a winter set. There are usually a few (very few) track worthy tires that also have good street manners, meaning not crazy loud, still grip fine when "cold" (above freezing of course), and decent wear (tread life) on the street. Then your winter set can be allseasons or "allweather" or "performance winter" or straight up snow+ice tires, depending on your winter driving/weather/preferences.
The other option would be a track set, and an allseason set for year-round street use (e.g. the DWS06+ or PSAS4 like you are planning). This gives much more flexibily with the track tires, to not care about street manners, but then you don't get to run real summer tires or real snow tires for your street use (unless you go to 3 sets of wheels).