Basically it amounts to how much of the building you want to do. Most of the parts can be purchased as kits or assemblies, reducing some of the small parts headaches.
Here's a few methods to build a fairly standard 5.56 caliber AR15 with iron sights:
1) Buy a complete lower half (i.e. stock receiver and trigger) and a complete upper half (receiver barrel and internals.) Then just push two pins together.
2) Buy a Stripped lower receiver (it's the serialed part that has to go through an FFL) and a "kit." The kit will include everything you need to finish your lower receiver. There's a group buy on another board for the kit that is 600 - 650 bucks shipped. Be warned, there is a huge backlog, something like 22 weeks. The upper half will already be assembled for you. You'll have to install the stock, grip, trigger and small pins and springs into your stipped lower receiver. It's a great learning experience while saving some of the more labor intensive tasks for another project.
3.) Build it all yourself. This will be one of the more difficult / time-consuming processes given the current parts availability.
I won't break it down to the individual pins and springs, simply because it isn't cost effective to source/buy each part one by one. Here's a rundown on parts:
First the Lower Half:
Stripped Lower receiver as explained above.
Lower Parts Kit (trigger, pins, springs, standard pistol grip)
Stock (Fixed or Telescoping)
Receiver End Plate, Buffer Tube, Buffer and Buffer Spring (Usually can be purchased with the stock as an assembly to save money)
Upper Half: (There's more than one way to skin a cat, but these are the basics.)
Upper Receiver Assembly: (Carry Handle or Flat-top) I'd recommend a complete assembly here, especially if you want to build a Carry Handle version as it will have the rear sight included/installed.
Bolt Carrier Assembly: (Best to buy it complete with bolt, as sourcing small parts is a PITA)
Barrel: If you aren't looking to spend a ton of money on tactical stuff like railed handguards for adding lights lasers, etc., an assembly is great here as well. It will usually include the barrel nut, gas tube and pin, the front sight or gas block(see below), standard handguards w/cap and delta ring assembly, and flash hider. The barrel nut, gas block/front sight and flash hider should arrive pre-installed if you go this route.
Sights: If you decide to build an AR15 with a carry handle and a Front Sight on your barrel, you won't need anything here to complete the job. If you buy a Flat-top upper, you will need a Rear Iron Sight. Flip-up models offer versatility if you mount a scope later on. Fixed Rear Sights are also available. If your barrel has a gas block, and you want to use standard iron sights, you'll need to make sure your gas block has a rail on it. Otherwise, you're getting into more expensive options like the railed handguard I mentioned earlier.
Magazines: Plural isn't optional here. Stock up.
Ammo: Hopefully, you stocked up when stuff like PMC ammo was $3.50 per 20 or cheaper. I did not.
TOMCAT, Thanks for such an in depth reply, I am going to print out and save this for when I get started later. So far, I know I want it to be a 223 (I have reloading equipment coming from Longtooth that has 223 dies), I also am going to want to put a scope on it. I was researching and am not quite sure which is better, the gas tube or blowback...Or I may have read that completely wrong. Anyways, your reply was extremely helpful, Thanks, Skip