Ok, first, if you handle multi-classing as it is in the PHB, the 4th printing has some specific rules. First, as Galannor points out, that decision is made at the time of character creation, but since your players are new to C&C and still first level, you can allow a minor edit to grant that to the player. I will point out the armor restrictions below. If you want to use a 3rd. Ed. style of multi-classing, this can work, you can simply allow the character to switch to the new class, and raise it's XP instead of the original class. This can work in a couple of ways, the first is to say that any xp gained from more combat related rewards goes to the combat oriented class, while any spell casting awards go to the spell casting class, other awards are divided equally between them. Not a very practical solution, but it could work. Another way to do it is only allow the switch at the time the character levels, so once he reaches 2nd level of assassin, he can add the illusionist, and gain xp only as an illusionist, at the next level, he can switch back or stay the same. You still get to use the abilities of both (with restrictions as a standard multi-class as shown below) but only gain xp to one class at a time. (This is my least favorite option) Finally you could treat it in the same way as Original D&D did with the Elf. Each adventure the player decides to to have the character act either as an assassin or an illusionist. If during that adventure the character uses abilities from the other class, they gain only 1/2 xp for the adventure, or perhaps if you think this is to strict, do this per game session, if very generous, only for that encounter. To do this you will need to make a unified XP chart, I would take 100% of the XP from the chart that has a larger value, and 1/4 the xp of the other class. That said, I am not really a fan of any of these options, but they are all workable.
I would allow the edit of the character to include either the traditional multi-class, requiring that the character split XP between the two classes, and level each up separately, or the class and a half system. And really, the class and a half system would work seemlessly for the character right now, since at 1st level you do not have the abilities of the supporting class, it is considered level-0, you do gain some of the functions, armor and weapon selection for example, but you do not get BtH, spells, etc. (though you could allow a couple of cantrips, perhaps based on the Int. bonus for a spell casting supporting class at character level 1). Starting at 2nd level the character would be A-2/Ill-1, then gain a level as Illusionist at every even level thereafter.
Armor restrictions are listed as follows-
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Multi-class
Multi-class characters can use any armor from any of the combined classes’ armor allowed list at however, they still suffer any penalties mentioned for a class ability such as with the rogue’s pick pocket.
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Class and a half
Wizard or illusionist supported by an armor proficient class may cast spells while armored, however, any spells which allow a save are granted a bonus to the save equal to the base AC bonus of the spell-caster’s armor. This
principle also applies to druids who wear metal armor. Wizard or illusionist supporting an armor proficient class cannot cast spells which directly cause damage, or allow a save, while wearing armor. This
principle applies to druids wearing metal armor.
Thus, mechanically speaking, in traditional multi-class, no armor may be worn if you are casting spells. however, in class and a half, with this character being an assassin supported by an illusionist, he could cast spells while armored, but only spells that do not deal direct damage. Since most illusions do not this makes assassin/illusionist a very powerful combination, but spells that do cause direct damage like the various Dark Chaos, Dragon Bite, Dragon Breath, etc. would not be able to be cast. Note that I break both of these rules if Elven Chainmail is acquired and used, and with some, exceptionally rare, magic armors.