First of all- welcome to the family!
The short answer? Yokohama ADVAN sports, stock size up front, 275 rear width.
Now the answer you want: The more disciplined you are about safe driving, the better tire you can get. Here's the thing: these cars have WAY more HP than the stock rear tires can handle, especially the base rims/tires. You'll fry the rears
before you've used 20% of the fronts and this equation is
extremely dangerous when the streets are wet. BECAUSE we don't get much rain here when we do it gets VERY greasy because of the oil buildup and traction goes from 100% to about 20% instantly.
That being said, keeping the stock 18's gives you more good options in tires and less cost per tire than 19" rims, so the important factors are a) your income level b) intentions with the car c) driving habits.
Generally the inexpensive tires have relatively poor or unpredictable grip, and are best for gentle mature drivers or students on a budget. Hankook Ventus V-12 is an example, or anything inexpensive from Falken or Kumho. The more expensive Grippier (BStone RE-11, Michelin Pilots) tires wear very fast (rears) but are predictable under hard cornering and braking and in my opinion worth the extra couple hundred dollars.
Keeping the 18's for a while? Know how to short shift and leave a gap when it's (even slightly) wet? Get the widest tires your 8" fronts and 9" rears will handle. Stay with the stock Yokos or Bridgestones the car came with, go wider.
Getting rims right away? Your avatar looks like you want to have some flush fit 19's or 20's with spacers and a 1" drop, which is fine. Price out tires to fit them first. Assume $100 install if you get them from Tire Rack or have them shipped to you. Get 19's with 9" fronts (for ease of turning) and 10-10.5" rears. Spacers are $100-$200 used
here.
Thinking of going on lots of canyon runs? Tracking your Z?
No room for error, and glad to hear you have cash to burn. Pilot Sport or RE-11's. Expert driver? Hankook RS-3 or Nitto NT01 (racing compound) and be SUPER careful in the rain.
Super tight budget? Just need rear tires for now?
If they're bald, don't wait, buy a pair of rears to match whatever fronts you have. It takes FOREVER to sell a set of base front tires here. FOREVER. The stock ADVANS are great tires. Go wider than stock on the rear! More meat = they will last longer and break loose less when you slam shift into 2nd.. Repeatedly.
Enjoy the ride!