Is my plan for adding a sub-panel reasonable?
You are severely under-sizing your cables/wires. Depending upon cable, independent conductors in conduit, over the air, etc. considerations.
For a starter your 100 amp feed to the first sub-panel should be 1 gauge copper, or 2 gauge if THHN/THWN.   The feed to the 50 amp sub-panel should at least be 6 gauge.
In your shed, 14/2 is fine on 15 amp breakers for lighting. Easier to work with than 12-2.
On both your planned sub-panels, you'll need to float the neutral (isolate from ground) and have 4-wire (2 hots, neutral and ground) to them.  In the separate outbuilding (shed) you'll need to also float the neutral and install ground rods and connect them to the grounding bus bar in the sub-panel.  You don't need to do this in a sub-panel in the same structure as the main panel, but you still need to isolate the grounds from the neutral.
Ampacity chart search results
In response to the comments:
It doesn't sound like you've done much load planning.   What are you planning on running in the shed, as well as on the sub-panel?   I would power the shed's sub-panel from the main panel, even if that means moving a couple more circuits to the sub-panel.   Also, 6 ga aluminum wire is considerably less expensive than 8 ga copper. Depending upon how long the run to the shed is, you might want to consider that. And yes, 40 amps on 8 ga copper or 6 ga aluminum is fine.   To connect the 100 amp sub, I'd go with a short length of conduit and individual wires. Depending upon your current configuration (in wall or surface mounted), you may need to &quot;assemble as you go&quot; because it would be difficult or impossible to install the conduit later.
You don't need to &quot;buy 4 wires&quot;, buy 1 length of 1 ga copper, 3 times the length you need and cut it as you install it.  Code tape (wrap) the neutral with white tape. A 6 ga green neutral is fine for ground. Alternatively, I THINK if you ran metal (EMT or rigid) between the main and sub, that would serve as the ground connection. (not sure!)
Also, others may chime in here, but panels are pretty inexpensive compared to the work involved in have to swap them out later if too small, so go big!
Lastly, and please don't take offense to this, but you are asking some pretty basic questions and initially made a pretty bad and dangerous proposal. Electricity is pretty dangerous if not handled properly.  If you still want to DIY, it would be a good idea to find a buddy that knows electrical well or even hire an electrician to check your work or get his advice before work is started.
I'm not a licensed electrican, but know quite a bit about it and the code and have done a lot of wiring.   Even so, when I wired my son's new house, I paid a licensed electrician to do a &quot;pre-inspection&quot; in case I missed something.  And he did find a few things that would have been called out in an inspection.
Hope this helps.  There is no honor lost in getting help / advice from pros.  In your case, you might need to do that locally.