I want my guy to be mostly a strength-based swordfighter, but I'd like to be able to sneak attack and possibly sneak/hide too. Would having medium armor and/or a shield hamstring that?
I'm not the most knowledgeable guy when it comes to character building. But from what I know:
The heavier your armor and shield are, the higher your armor check penalty and the lower your max DEX bonus (in the case of armor). A low max DEX bonus would make no difference in the case of a STR rogue, as you wouldn't have enough DEX bonus to reach the cap anyway. But if you invest in STR instead of DEX, you will already be worse at stealthy stuff compared to a DEX rogue since you'd hardly get any DEX bonus to your skills, and armor and shield would additionally give you penalties, further reducing your chances of success. Combine that with multiclassing, which means you will have less skillpoints to invest in rogue abilities, and I think it's likely that most creatures that more or less match your level would spot you trying to sneak up on them. Example for DEX bonus and penalties:
DEX 12 (+1)
Medium armor AC4, that is chain shirt or scale mail (-2) or Medium armor AC6, that is chainmail or breastplate (-5)
Small shield AC+1 (-1), or Large Shield AC+2 (-2), or Tower Shield AC+3 (-10)
So a fighter/rogue with DEX 12, lighter medium armor and a small shield would only get -2 to skill checks like sneaking (which might still work if your sneaking skills are high, but as multiclassing STR-based character they probably won't be), whereas a fighter/rogue with DEX12, stronger medium armor and large shield would get -6 already. If you try to sneak in full plate (-8) with tower shield, it would be -17 even, so a very low chance to stay undetected.
That being said, Knockdown would be a useful fighter feat for you to take, as knocked down enemies are considered "flat-footed" and as such vulnerable to sneak attacks even if you're not stealthy. And you also get sneak attacks on enemies that are distracted by fighting other targets, so any distractions, e.g. by henchmen, would work in your favor, too.
An alternative to multiclassing would be ranger as sneaky fighter class, because rangers can hide and move silently, too, but they are worse fighters and don't get the rogue's sneak attack.