I am finishing out my 1000 sq. ft. basement as an apartment. The main panel is 200A and is full (all electric house. I will need another 200A panel in the basement for all the electrical needs, including range, dryer.
Can I just simply tap into the incoming power lines on the main panel with 6-3 and drop it down to the new panel in the basement?

You really need to start this conversation with the power company to determine how they are willing to increase the power to your home. Next, call the local authority having jurisdiction and explain what you want to do. Then you need pull a permit to do the work.

If you have an licensed electrician doing the work, make sure that the electrician pulls the permit in his/her name. That way, if the inspector finds problems, the electrician is required to fix them because he pulled the permit. If the permit is in your name, then you are required to have the error corrected.

The NEC requires a minimum service of 100 amps, so you may not have to drop a full 200 amps to the apartment.

You cannot tap onto the main service lugs to feed a second panel. Flat out, no!

Also, I seriously doubt that your 200 amp main service panel will support a 200 amp breaker as a subpanel feeder. You need to check with the manufacturer of the current panelboard.

You really need a licensed electrician to do a load study for you.

Next your feeder is to small. If you try to feed a sub panel from your main service, you will need a #3 AWG Copper for 100 amps.

If you try to run a second 200 amp panel, you will need a 3/0 copper conductors.

I can probably get by with 100A.
You cannot tap onto the main service lugs to feed a second panel. So then where does it tap into?
I think it's time to call my buddy that helped me wire the house 20 yrs ago. I was working for him at the time. As a commercial electrician, he can pull the permit.

DonF, I am confused and it may be terminology, since I haven't worked as a commercial electrician for 20 years.
You posted this comment: A) have a four wire feed (2) Hots - (1) Neutral and (1) ground. from the Main Service Panel to the subpanel in the garage.
But above, you said not to tap into the main for the sub.

Since you worked as an electrician, can you tell me what would protect the #6-3 feeder from overcurrent or short circuit condition if you were to tap into the Main?

Keep in mind that a #6 copper wire, at best, can handle 60 amps, and the Main is 200 amps.

TK, I can not answer that. Since I haven't run any wire, I can run #3 for 100A. Just don't know where to run it to. Don't have much residential experience. It was all hotels and businesses. Never had to deal with subpanels. And it was 20 yrs ago.
I am trying to get a feasibility study here before I start. If it can't be done or will be too expensive, the dryer and stove may have to give way to gas. Which is a whole new can of worms.
I actually thought it wouldn't be difficult to add a sub to the main, but you guys that know this stuff are making me question the practicality.

From the main service panel, use a DPDT breaker and feed a four wire feed.

Black to one side of the breaker, Red to the other side of the breaker, Neutral to the Neutral bus and Ground to the Ground bus.