Base attack bonus, skills, saves, and hit points all stack across the classes that you take. Spellcasting is one of the few things that was really kludgy in 3E and 3.5E D&D as far as stacking. They needed the Practiced Spellcaster feat, Mystic Theurge, Spellsword, Eldritch Knight, etc. I'd like to suggest that including some mechanics (could be an optional rule) for stacking of spellcaster level would be a good way to differentiate from 4E D&D and show that multi-classing can be factored in relatively cleanly to the rules. Unearthed Arcana had an optional rule for this (page 136, Table 5-1).
One question is whether divine and arcane spellcaster levels should stack? If they don't, you're favoring say Paladin/Cleric builds versus Paladin/Sorcerer for example.
Another question is whether non-spellcasting classes should get some limited progression? If say a Rogue gets one spellcaster level per four levels, then a Rogue 16/Wizard 1 has a spellcaster level of 5 - not great, but at least those 1st level Wizard spells make a bit more sense and last more than a round.
Stacked spellcaster level should not affect access to spells/day, only caster level. So if say arcane and divine spellcaster levels stack, a Cleric 10/Wizard 10 casts all spells as a 20th level caster, but still only has access to spells/day and spell levels as a Cleric 10 and Wizard 10.
Thoughts?
One other suggestion here, give Bard the Arcane Armor Training feat instead of their ability to cast spells in light armor with shield. This way they can choose to work up to heavier armor if they want (say a Fighter/Bard) by taking Arcane Armor Mastery. Likewise, a Bard/Wizard would then make a little more sense as you can cast those other arcane spells with reduced arcane spell failure too.