There are quite a lot of options for this.
I’m going to start with the simple stuff since you asked a pretty basic question, but I’m also going to expand that into a fair few options that are generally a bit better.
When your wife hits 5th-level, she can take a level of a spellcasting class, e.g. Wizard. Bam, she now has spells.
You calculate her stats by adding what a Wizard 1 gets to what she already has by virtue of being a Fighter 4. That is, she gets d4+Con HP, +2 to Will, 2+Int skill points, the ability to summon a Familiar, the Scribe Scroll feat, the opportunity to specialize in a school of magic, and a number of 0th- and 1st-level spells each day, including bonus spells for a high Intelligence score.
There is a complication here in that “a wizard begins play with a spellbook [with some spells already in it],” which is sort of ambiguous about what happens when you didn’t start as a Wizard but rather multiclass into the class. I suggest that you give her the spellbook she’d be entitled to shortly after becoming a Wizard, or perhaps as part of the process by which she starts her wizarding career. This is a fine time to get a spellbook. Certainly don’t send her into combat with those stats and no spells, and expect her to be able to fight like a 5th-level character, though.
The rules explicitly and intentionally do not describe exactly how one goes from one level to another, so you are free to describe this however makes the most sense (finally reading up on magic like you promised Ma you would, enrolling in an arcane university, becoming an apprentice, whatever).
Obviously, if she chooses a different spellcasting class (aside from Complete Arcane’s Wu Jen, who also uses a spellbook, or the Archivist from Heroes of Horror who uses a “prayerbook”), there is no need to worry about this.
Multiclassing and spellcasting
Basically, spellcasting levels don’t play very nicely with levels in other classes. Where melee characters get things like BAB, which all classes grant, and bonus feats, which you typically qualify for based on your stats rather than your levels, spellcasting is purely level-based. A Fighter 19/Wizard 1 has no better spells than a Wizard 1 does. At 5th-level, when a Wizard 5 has 3rd-level spells, a handful of 0th- and 1st-level spells is not quite as useful. Still useful, no doubt, but not really quite as useful as you’d expect to get at 5th level.
There are some solutions to this. Complete Arcane provides the Practiced Spellcaster feat, which provides a bonus to Caster Level up to +4, but is limited by your HD. This means that you can “make up” for the Fighter levels a little bit. You don’t get higher-level or more spells, but at least the spells you have will have the potency they would if cast by a higher-level spellcaster.
Arcane-specific problems: multiclassing and melee fighting
The Sorcerer and Wizard classes have poor BAB and HD, which isn’t good if you want to continue fighting in melee.
Divine spellcasting classes and psionic manifesting classes have much less difficulty here. Neither is affected by Arcane Spell Failure, which is a fairly big problem for a Fighter since armor proficiencies are a major Fighter class feature. Cleric and Druid both have ¾ BAB, which is usually pretty good, and Clerics even get divine power which gives them full BAB as long as it lasts. On the psionic side, Ardents (Complete Psionics) and Psychic Warriors get ¾ BAB also, and the War Mind prestige class is great since it gets full BAB and progresses faster than Psychic Warrior.
But you don’t have to give up on arcane spellcasting. There are many options for mitigating Arcane Spell Failure and Prestige Classes can have better martial stats. The Eldritch Knight from the Dungeon Master’s Guide has full BAB and a higher HD, though you lose another spellcasting level, which is very painful, particularly since you have left the realm where Practiced Spellcaster can help you.
Better options include Abjurant Champion (Complete Mage) and, perfect for a Dwarf, the excellent Runesmith (Races of Stone). If you have this book, I cannot stress enough how good Runesmith is here for your wife’s character. It allows her to cast her spells while wearing armor, gives her much better stats than Wizard otherwise would, and avoids more lost spellcasting levels. Plus fun stuff like giving other characters “runes” that they can use to cast some of her spells themselves.
I therefore recommend Fighter 4/Wizard 1/Runesmith 5/Abjurant Champion 5. The other 5 levels, if she gets that far, are harder, and I’ll have to revisit this answer if you need suggestions for them. Alternatively, Fighter 4/Cleric 16 (or Fighter 4/Cleric-and-some-Prestige-Class 16) works quite well, as does Fighter 4/Psychic Warrior 16 or Fighter 4/something 1/War Mind 10.