This isn't meant to be a guide on what order you should pick, but more of a post on how the order of classes can affect your character's stats. The caps are based on being 9x40 (max level) and should be used to plan out your character. As you're leveling the caps for some things will be lower and will increase every time you gain a level 40.
Warrior
This will influence your weapon damage cap, which applies a bonus to your autoattack. 1 weapon damage = 1 point of damage done, does not work with some weapons equipped such as daggers. You will also get more hps per level if this is higher up in your order, but I don't know what values that would be.
1: 25 weapon damage
2: 23 weapon damage
3: 21 weapon damage
4: 19 weapon damage
5: 15 weapon damage
6: 13 weapon damage
7
8
Thief
This will influence your stab damage cap, and I believe thief prime gets some extra damage on their backstab. Prime thieves will also have circle damage cap, while anything outside of thief prime will just be 0.
1: 67 stab damage, 75 circle damage
2
3
4
5
6
7
8: 32 stab damage
Monk
This will affect your hand damage and damage reduction caps. Monk prime also seems to have a bonus modifier for strike damage.
1: 29 damage reduction, 33 hand damage
2: 25 damage reduction
3
4
5: 17 damage reduction
6: 15 damage reduction, 23 hand damage
7: 13 damage reduction
8: 11 damage reduction
Bard
This should affect your charisma cap, but I don't think anyone outside of bard prime would make use of charisma on a normal basis.
1: 28 charisma
2
3
4
5
6: 16 charisma
7: 14 charisma
8
Mage
This will affect your saves and spell bonus caps.
1: -31 saves
2: -25 saves
3: -23 saves, 92 spell bonus
4
5
6
7
8
Cleric
This will affect your heal bonus cap. Also class order reflects what you can summon with supplication, which won't be listed here since it's clear in the help files on what class order does with that spell.
1
2
3
4: 77 heal bonus
5: 67 heal bonus
6
7
8
Druid
I don't think druid has any specific caps tied to it, but they tend to get more mana the higher in your class order and there's some sort of bonus mana regen that is applied to prime and secondary. The exact value of the regen isn't known. Druid class order also affects what forms you can shift into, but the help file in game is clear on this.
Necromancer
This will affect your undead control cap. It also affects what creatures you can summon, but the help files in game are clear on this. Generally prime/secondary can get swordwraiths, 4th can get phantoms, and beyond that the pets would be too weak to be useful at high levels.
1: 27 undead control
2
3
4: 19 undead control
5
6
7
8: 11 undead control
I'll fill this in as I get more data from players and my other characters, hope this helps on making the choice on where you want to place something. For anyone wanting to contribute for this you can view the values using msdp at msdp.CHARACTER_STATISTICS.BONUSES. If you can provide the value listed under CAP and your class order I can fill this out more.