The ability to read and cast spells is not restricted by the new form (Planetar).
- To answer your second question first: Shapechange explicitly does not permit use of the new form's class levels or spell casting traits (the wizard would not accrue the innate spell casting feature of the Planetar) but the wizard
will retain the benefit of any features from your class, race or other
source, and can use them provided that your new form is physically
capable of doing so. (PHB p. 274-275)
The wizard can thus cast his spells. An angel/Planetar is able to cast a spell and to read. (It might be wise to hide an arcane focus with the spell book to avoid the problem of spell pouch/focus changing as well). The Planetar's spell casting without components is linked to its innate spell casting.
The planetar's spellcasting ability is Charisma (spell save DC 20).
The planetar can innately cast the following spells, requiring no
material components ...(MM. p.17)
- In the True Polymorph case, the rules don't expressly forbid spells
in the new form, nor do they state that prepared spells are lost,
nor that they are replaced with the new form's spell casting
feature. Your DM will have to rule on whether or not the Innate
Spell casting feature of the Planetar, or the Arcane prepared spell
casting class feature of the Wizard, will be operative while in the new
form. It would be reasonable to rule that the wizard's prepared spells to remain since the polymorphed creature
retains its personality and alignment. (PHB p. 283)
... and who you are is a wizard. If the ruling was that the new form has innate spell casting of a Planetar, it would make sense that this replaced the prepared/Arcane spell casting of the wizard. (At that level, this could represent a heck of a loss if early in the spell casting day, but a real bonus if most other spells have been expended ...) The replacement of the mental statistics does not, per the spell language, necessarily change "who you are" while transformed under this spell.
... can't speak, cast spells, or take any other action that requires
hands or speech unless its new form is capable of such actions.
The Planetar is capable of such actions.
Who is really in there? The Prince turned into a frog, or is that just a frog?
Note: if the wizard and a Planetar are not the same alignment (LG) this could pose another problem (that probably needs its own question). A simple approach would be that the Planetar's spells are not available -- but that depends on how your DM works with alignment at your table. Arcane spells, and the ability to use them, are not alignment dependent. The Planetar's spells, as they are innate to that creature, appear to be alignment dependent.
As angels are "formed of the divine essence of benevolent gods ...
they are "the embodiment of law and good" (M p. 15)