Iam installing a 100 amp sub panel in my basement. The new panel is right beside the 200 amp panel. I am installing 2 20 am circuits and 5 15 amp circuits. What size wire do i need for this installation and how do i connect it to the 200 amp panel?

Your going about this backwards.

First you do a load calculation and then you know how much amperage needs to delivered to the panel.

Once you know the load the panel will have to carry then you can plan you feed and branch circuits.