Once you have received your pistol permit (if you don't have your permit yet, go here), you can go out and buy your first firearm!  Congrats on making it to this step!

Federal Law states you may only buy a handgun in the state in which you reside.

You can only buy a handgun in Connecticut, if in addition to being a resident, you have a valid Permit to carry Pistols or Revolvers, a valid Eligibility Certificate, if you are a licensed Firearms Dealer or if you are a Sworn Police Officer (see the CT State Police Special Licensing and Firearms Unit for more specific information).

Since you've taken the NRA Basic Pistol course, you should have enough info to make you dangerous in deciding on what you should purchase as your first firearm.  

Now you can to your local gun store (a quick google search for "gun stores" should help you find one) OR you can actually purchase a firearm online at larger chain retailers.

Even though you can purchase a firearm online, it WILL NOT be delivered to your door.  There are two forms, Federal (Form 4473) and State (DPS-3-C), which you need to fill out.  Additionally, the store needs to verify you have a valid pistol permit.

Once you complete the paperwork, the store will run a NICS check and will obtain a firearms sales authorization number through the CT State Police Special Licensing and Firearms Unit.

Assuming those steps are completed successfully, you will be able to walk out of the store with your brand new firearm.

*Note:  The paperwork process can take around 30-40 minutes.  Depending on how busy the store is, etc, etc.  But if this is your first gun, give yourself plenty of time to make this purchase.