This combination has the same tradeoffs and dynamics as most martial/ spellcasting combinations, but the variability of the fighter class helps somewhat. Because fighters can use their Strength or their Dexterity as their weapon-using ability score and bards only need Charisma to multiclass, there’s considerable freedom to build such a character how you want to.
The flavor, while not exactly complementary, doesn’t exactly contrast, either. The fighter class is, by design, something of a blank slate, able to cover a wide variety of different combat styles and warrior types, and it’s therefore relatively easy to “match” a fighter archetype to the chosen bard one.
Focusing more on bard levels will give you your standard clever, tricky bard who is noticeably better at just straight-up fighting than is normal for a bard, such as Vahli from The Coward. More levels in fighter will give you a dedicated warrior who has picked up a touch of magic and showmanship along the way. An even mix of both makes for an unusually-magnetic fighting spellcaster who can mix potent spellcasting and competent weapon skills on the fly.
[[Synergy Feats for Multiclassers#Bard w/ Fighter|Bard-Fighter chains and ideas]]