I read "tighten up the handling" and I feel like pontificating so here it goes.
If banging out the best laptime is your priority you have two primary objectives: maximize traction, increase consistency.
I'm kind of assuming you'll be driving at 10/10ths and not just dodging potholes but I've got a halfway option and a ballsout option.
1. Get the wheels pointed in the proper direction.
A quality alignment at a specialty shop (not Sears or some chain) along with a bare minimum of camber bolts to get the most camber on the front wheels as possible, -1 degree should be a minimum. While lowering car nets you more camber a quality setup may not fit in the budget (entry level setups are $2k-$3k at this point, nobody has a proven budget setup to my knowledge, some good ones but no amazing ones for <$2k) and you may not be even left to right. If you want to go all out camber plates so you can get more than -2 degrees, from what I've seen most people aren't running much more than -3 degrees. Toe can massively affect the feel and agility, zero toe front, a little bit toe-in rear is common for agile "low powered" rwd sports cars, it's stable and responsive.
You can blow thousands of dollars on wheels and tires but what's the point if you're only using ~70% or less of the rubber you just shot your wad on because the tires aren't pointed in the right direction?
$200-$400 for front camber bolts + alignment
$700-$900 for front camber plates + alignment
+~$500 rear lower control arms
2. Brakes.
This is not a big deal if you're doing casual weekend stuff, but even light autocross or a steep windy downhill will exceed the limits of the stock pads and fluid. Without consistent and reliable brakes you will second guess the car and waste valuable seat time preserving equipment instead of driving it and exploring your limits. A hybrid pad is truly a compromise but if you're not hitting the track or cones will probably be more than enough otherwise budget for a true race pad, plenty of threads for research on this topic. I boiled OEM fluid my second track day, again if you're just tooling around on the weekend this is probably overkill.
$200-$300 hybrid pads, ATE Type 200 brake fluid flush (or equivalent)
$400-$500 track pads, track fluid flush
3. Tires
Now I'd be cautious about going too fast with a rubber upgrade, imo put the money in the bank, burn up the stock rubber and learn how to deal with low grip, when you upgrade the tires the same principles and driving style will apply, you'll be doing it at higher speeds with less margin for error (both in terms of your actions and the repurcussions). Now, lighter wheels will definitely be an upgrade but jamming on the fattest rubber will likely be detrimental to laptimes with stock power as you've added drag on the straights.
~$500 for 4 sticky DD-able tires in OEM size mounted
I saw an amazing deal for $1,500 for Enkei RPF1's 17x9 with Dunlop Direzza ZII Star Specs (quality sticky tire, popular choice for autocross and amateur racing, DD capable if you don't deal with snow or lots of water). Typically those wheels would be around $1k, tires up around $700 I think. If kept a little smaller as suggested above the ~$1500 mark is about right for 17x8 (imo 18's are too big a diameter, I'd even consider going to 16 when I go to buy)
There you go, all the budget gone in three areas if you go balls out, these are the biggest improvements you can make when it comes to raw speed in any scenario that isn't a drag strip. These are the biggest deficiencies in the car as it comes from the factory if your intention is to AutoX or HPDE or even bombing down a canyon road.
This is the path I've gone down, increasing camber is next for me as the camber bolts weren't enough, while some of the smaller things like bushings and braces such may help with the feel, the way the car is set up from the factory all those things are more than adequate as in 99% of owners will never see a proven improvement.
Since you don't mention performance driving I'd guess you'll stay on the low side with camber bolts and hybrid pads, along with wheels and tires what I suggest is about $2k. Next would be coilovers for handling by my guess, there are entry level sets like Bilstein with street setups that can be had within budget, lots of options and research to do.
And at the end of the day, putting a smile on your face is important, do whatever feels right, big ol' exhaust, pink wheels, those door latch upgrades that supposedly stiffen the whole chassis.
