WIZARDRY
- shares 11 spells with Alchemy, 9 with Psionics, and 6 with Divinity
- 9 spells are covered by instruments (2x lvl 6/7), 8 by gadgets (1x lvl 6/7)
- has 4 out-of-combat spells exclusively (Missile Shield, Enchanted Blade, Shadow Hound, X-Ray); only X-Ray can be covered by a gadget, none be instruments

DIVINITY
- shares 12 spells with Psionics, 11 with Alchemy, and 6 with Wizardry
- 8 spells are covered by instruments (2x lvl 6/7), 6 by gadgets (1x lvl 6/7)
- has 2 out-of-combat spells exclusively; one can be covered by a gadget (Armorplate), the other by an instrument (Magic Screen)

ALCHEMY
- shares 11 spells with Wizardry, 11 with Divinity, and 6 with Psionics
- 1 spell is covered by an instrument, 6 spells by gadgets (3x lvl 6/7)

PSIONICS
- shares 12 spells with Divinity, 9 with Wizardry, and 6 with Alchemy
- 14 spells are covered by instruments (3x lvl 6/7), 6 by gadgets (1x lvl 6/7)

INSTRUMENTS AND GADGETS
- share only 2 spells (Terror, Heal All)

SOME CONCLUSIONS:
- It is not unreasonably to have both a Gadgeteer and a Bard in your party, since they share only 2 spells and peak at different times in the game.
- If you have a Bard you probably don't need a Psionic (14 Psionic spells can be covered by instruments).
- Gadgets are more evenly spread in terms of spell schools, but for high levels there is a stronger focus on Alchemy (3 of the 7 level 6/7 Alchemy spells can be covered by gadgets).
- In a party with a Bard and no Gadgeteer, if you need to decide on one specialized caster, the best choice is probably the Alchemist (they share only 1 spell: Sleep)
- In case you intend to dual-class specialized spell casters (i.e. no Bishop), you're probably best off by starting with a Mage or Priest (for their exclusive level 3 buffs) and change to a Psionic or (usually preferrably) Alchemist by character level 6 (to do that you might need to add points to SEN or DEX, respectively).


EDIT: Typo corrected

Here are some other thoughts I have:

First, there are a few spells I consider essential; in other words, every party *should* have the means to cast it:
Heal Wounds
Identify (note that Identify Scrolls only give you level 4)
Missile Shield
Enchanted Blade
Armorplate
Magic Screen

There are also some very useful spells not on the list, but I put these on because the game basically assumes you have it.

Notice that every one of those spells can be used by a Priest or Mage. Note that Vi Domina can replace the Priest, provided you are willing to drag her into areas she doesn't normally go.

Alternatively, you can replace the Priest with a combination of a Bard (for Magic Screen), a Gadgeteer (for Armorplate), and a Psionic (because you still need Identify).

One other note is that some spells can be covered with items. The Amulet of Healing, for example, casts a powerful Heal All spell and can be recharged by selling to Lord Braffit and buying it back. It's expensive to purchase them from Crook, but there are no less than 2 that can be found. (Note that they also boost AC by 3, which makes them so powerful that I have a self-imposed rule of not giving more than one to the same character.) The much cheaper Amulet of Life allows Resurrection to be covered cheaply, provided you have someone with 35 Artifacts.

Dual-classing spellcasters is *not* recommended; the mechanics do not favor that choice. Either go from caster to non-caster (and accept limited spellcasting ability and that your spells will not work on enemies), or from pure (like Priest) to hybrid (like Lord), or for a specific skill (like Stealth or Critical Strike).

Also, one mistake: You write that Psionics shares 6 spells with Psionics; do you mean that it shares them with Alchemy instead?

Thanks, changed that.

As for recommendation of dual-classing spellcasters: What you say is very true for full parties (6 characters), but it might be worthwile in smaller parties where you have trouble covering all there is to cover and you level up faster.

For example, a 3-headed party with:
1 Lord/Valkyrie (maybe class-changed from a Priest at lvl 2),
1 Bard (possibly class-changed to a Fighter or Rogue once Music is fully develped)
1 Mage > Alchemist (class-changed at level 4)

You could even think about wating out until level 9 and pick up stuff like Freeze Flesh (maybe not essential, since the Bard gets it too), Fireball, and especially X-Ray, along with shared spells like Element Shield, Noxious Fumes, Crush etc.

But sure, for normal parties, changing through different spell schools is usually not the way to go.

The problem is that Bishop is always the better choice. Instead of Mage 3 > Alchemist, you use Bishop 3 > alchemist, you get, with minimal experience cost:
More HP
3 more Alchemist caster levels
The possibility of learning other types of magic if desired.

Even at higher levels, a Bishop will be at most 2 levels behind a pure caster. A pure caster multi-class will be further behind if you want any spells of second level or higher. The only exception is spells in multiple schools, but high level spells tend to be single school. (Note that every 7th level spell is single school.)

If you want multiple spellbooks on one character, you want a Bishop, not a multi-class character, even if you only want two spellbooks.

One question on Bards/Gadgeteers:
Does the class level vs. monster level dice roll also apply to instruments and gadgets?
I would thinks so, which would be a serious argument against the common level 19+ class change of Bard to Rogue or Fighter. I guess it's not THAT bad since many good Bard instruments are party buffs that require no level comparison, but stuff like the Arresting Aria, Banshee's Howl, Chaos Drone, or Strings of Seduction probably won't work as well late game if one choses to class change.

Yes, I am pretty sure it does apply.

Also, your level also limits the power level at which a spell can be reliably be cast. For instance, at level 18, a 7th level spell can only be reliably cast at power level 1, regardless of the corresponding skill. If you want to be able to reliably use level 7 spells at power level 7, you need to be at least level 24 to cast it without risk of fizzle or backfire. (Also, note that you also need more than 100 skill for this.)

For Bards this is somewhat mitigated, since they reasonably soon already 'cast' at the highest level, but if spell resistance plays a roll (and if it's "Bard level" that matters and not "Character level"), then we should think twice about class changing our Bard. Especially if the musical spells play a signifant roll in our combat approach.

I assume you want to say Alchemy shares 11 with Divinity (not Alchemy)?