If you can't use tandem breakers then a sub-panel is in order. SquareD and Cutler-Hammer CH panels are generally well liked. I'd probably install something with 8-10 spaces served with a 60 amp breaker to accommodate any future needs. I'm assuming the sub-panel will be located in close proximity to the main service panel so the cost of the larger wire will be negligible.

I would use the same brand sub panel as the main panel you currently have. No sense in mixing brands. The breakers will then be usable in either panel.
You will need to move two circuits from the existing panel to make room for the breaker to feed the new sub panel.

Yep, you can not use tandems in the 4040.
As Joed points out, barring some bizarrely obsolete main panel (which the Siemens is NOT), I'd use another one of the same brand. I use SquareD to augment my Zinsco main panel, but I eventually came back and put a new squared main in to match.

I am moving forward and purchased the Seimens 125 amp subpanel, #2 THHN feedwire and 100 amp 2-pole circuit breaker. I seleted a large subpanel, because I plan to add a bathroom in the future too. My questions include: 1) Is there any issue if I install the subpanel below the main breaker panel then run the new wires into the top of the main breaker panel, through the main breaker panel, then into the top of the subpanel? This would prevent seeing the wires on the outside of the subpanel. 2) Also, is there a minimum distance from the basement floor to install the subpanel?

One thing to remember, you will have to run 4 wires from the main panel to the sub-panel. The only place you can tie neutral and ground together is at the main panel.

There's no restriction on panel placement above the floor. As long as you can get the cover off/open and you have the clear area in front of it.

As for running all the new circuits through the main panel. I hope you have enough knockouts to do all that.

Putting it below may get interpreted as being in the main panel's dedicated space. Some inspectors have not even allowed a receptacle to mounted directly below a panel. I think it would be sloppy looking and depending on the height of your current panel, the sub would be a pain to work on. To pass through the main panel, you would need one or more conduit nipples between the 2 panels. Keep the neutral wires together with the hots so that the neutral lands in the sub panel.

I plan to run three #2 THHN feedwires from the main panel and the sub panel. Two of the wires will be hot. One will be neutral. As for the ground, what size should be used? Is a ground wire from a 14-2 large enough?

A 14 gauge ground feeding a 100 amp sub panel? heck no! (: You need a #6 green or bare wire. Make sure the ground bar is isolated from neutral in your new sub panel.

" Is a ground wire from a 14-2 large enough?"

This question scares me. :(

I appreciate the recommendations. Below are my next steps for your review and comment:

The subpanel will be mounted on the right side of the main panel. The distance between panels will be about 4 inches. The subpanel will be about 5 feet from the floor. In other words, I will not be installing the subpanel below the main panel as previously considered.

As recommended, I purchased #6 Green Wire (6 AWG THHN) for grounding. It will be routed from the main panel ground bar to a new ground bar in the subpanel. This new ground bar was purchased separately because only one bar was in the new subpanel. The new ground bar kit is 14 position, #14-4 AWG. By adding the ground bar, the neutral will be isolated from the ground. No bonding strap will be installed on the ground bar.

The bar that came with the subpanel will be used for the neutral. I will attach a bonding strap to the neutral bar and screw the other end into the back of the subpanel.

The subfeed consisting of two hots, neutral, and ground will be routed from the main panel to the subpanel. This subfeed outside of the panels will be encased in 1-1/4" diameter flexible conduit. The subfeed will be routed from the side of the subpanel and into the side of the main panel. The conduit distance between the panels is short, i.e. 4 inches.

I plan to run the new circuits into the top of the subpanel.

Any additional recommendations are appreciated. Thanks.

The bar that came with the subpanel will be used for the neutral. I will attach a bonding strap to the neutral bar and screw the other end into the back of the subpanel.

I do believe you got this backwards. Keep the neutral separate and fasten the ground bar to the case. I bought a subpanel with the neutral and ground installed with a bar to ground the neutral if desired which I did discard. The ground was fastened to the case